# Rieger, Schwabe, Suess-de Vries: The Sunny Beats of Resonance

F. Stefani  G.M. Horstmann  M. Klevs  G. Mamatsashvili  T. Weier  Helmholtz-
Zentrum Dresden – Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany
University of Latvia, Institute for Numerical Modelling, 3 Jelgavas street,
Riga, LV-1004, Latvia

###### keywords:

Solar cycle, Models Helicity, Theory

##  1 Introduction

The goal of this paper is to sketch out a comprehensive model of the various
periodicities of the solar dynamo that appear on widely different time scales.
These include, on the high-frequency end, Rieger-type periods in the range of
a few hundred days, then - most prominently - the 11-year Schwabe cycle, and
finally the longer-term 200-year Suess-de Vries cycle. While modern solar
dynamo theory has at its disposal a broad suit of models to plausibly explain
those different periodicities (more or less) separately (Charbonneau, 2020),
we aim here at a self-consistent explanation that relies on various
gravitational influences of the orbiting planets on the Sun and its dynamo.
Key to this concept is the claim that the Schwabe cycle represents a clocked,
if noisy, process with a mean period of 11.07 years that is synchronized by
the three-planet spring-tide periods of the tidally dominant Venus-Earth-
Jupiter system. We hypothesize that the necessary deposition of tidal energy
is accomplished, on Rieger-type timescales, as a resonant excitation of
magneto-Rossby waves by the corresponding two-planet spring tides, mainly
during cycle maxima with strong toroidal field. Utilizing this deposited
energy, the synchronization of the Schwabe cycle relies then on a parametric
resonance of a rather conventional $\alpha-\Omega$-dynamo with the three-
planet beat period of 11.07 years between the underlying two-planet spring
tides.

On the low-frequency side, we will pursue the conceptionally similar idea that
the Suess-de Vries cycle emerges as a beat period of 193 years, this time
between the 22.14-year Hale cycle and some spin-orbit coupling mechanism
related to the rosette-shaped motion of the Sun around the barycenter of the
solar system which, in turn, is dominated by the 19.86-year periodic syzygies
of Jupiter and Saturn.

Finally, we will speculate about the appearance of the longest-time
variabilities which usually go under the notion of Eddy and Bray-Hallstatt
cycles. While reiterating the intrinsic tendency of the Suess-de Vries cycle
to undergo chaotic breakdowns into grand minima, we will also discuss the
possibility that some Bray-Hallstatt-type periodicity might appear by virtue
of a stochastic resonance phenomenon related to the of 2318-year period of the
Jupiter-Saturn-Uranus-Neptun system which is also visible in the barycentric
motion of the Sun.

Yet, before entering these complicated issues we have to give at least a
preliminary answer to one elementary question…

##  2 Problem? What problem?

The main problem with the problem at hand is that even its very existence is
not generally accepted. Au contraire, a couple of recent papers by Nataf
(2022), Weisshaar, Cameron and Schüssler (2023), and Biswas et al. (2023) have
vehemently refuted any claim for phase stability or clocking of the solar
cycle, by referring to various data sets from the last millennium.

Interestingly, neither of those authors have taken into account the
complementary indication for clocking stemming from the early Holocene.
Analyzing two different sets of algae-related data, Vos et al. (2004) had
provided remarkable evidence for phase stability over thousand years, with a
mean period of 11.04 years, which is - in view of the implied error margins -
barely distinguishable from the 11.07-year period mentioned above. A little
blemish of these data - the appearance of a few intervening phase jumps of 5.5
years - was plausibly attributed to the specific optimality conditions of
algae growth. In Stefani et al. (2020b) the signature of the arising “virtual”
phase jumps was also shown to be well distinguishable from that of any “real”
phase jumps the possible occurrence of which was several times discussed
(Link, 1978; Usoskin, Mursula and Kovaltsov, 2002) in connection with the
solar cycle during the last millennium.

As for the latter, things are even more complicated than for the early
Holocene period. While the series of solar cycles can be safely reconstructed
from telescopic data back to A.D. 1700, say, the cycle reconstruction during
the Maunder minimum is already quite questionable, and even more so in the
centuries before.

In frame of his “Spectrum of Time” project (Schove, 1955, 1983, 1984) the
British scientist D. Justin Schove had compiled an enormous body of
telescopic, naked eye, and aurora borealis observations into an uninterrupted
series of solar cycles going back to A.D. 242 (with some interruptions, even
back to 600 B.C.). Not surprisingly, though, this ambitious project has met
profound criticism, the gravest counter-argument referring to his “9 per
century” rule which automatically leads to a phase-stable solar cycle
(Usoskin, 2023; Nataf, 2022). While this argument is formally correct, it does
not refute the possibility that the solar cycle is indeed clocked with an
11.07-year period, in which case any auxiliary and occasional application of
the “9 per century” rule would do absolutely no harm to an otherwise correctly
inferred cycle series.

In this hardly resolvable situation, any reliable data sets that are
independent of Schove’s cycle reconstruction could play a decisive role and
would, therefore, be highly welcome. An interesting candidate for such a data
set, based on 14C data from tree-rings, was recently published by Brehm et al.
(2021) and further analyzed by Usoskin et al. (2021) who generated from them a
solar cycle series back to A.D. 970. While the ambiguities of this series were
clearly pointed out by Usoskin et al. (2021) (who carefully assigned various
quality flags to the individual cycle minima and maxima) it was uncritically
adopted in a recent publication by Weisshaar, Cameron and Schüssler (2023).
Applying statistical methods of Gough (1981) to distinguish between clocked
and random-walk processes, these authors claimed high statistical significance
for their finding of a non-clocked solar dynamo.

Yet, looking more deeply into the underlying cycle series of Usoskin et al.
(2021), Stefani, Beer, and Weier (2023a) identified at least one highly
suspicious cycle around 1845 for which there is no evidence from any other
cycle reconstruction. With this superfluous cycle being plausibly rejected, a
one-to-one match with Schove’s (phase-stable) cycle series was evidenced back
to (at least) 1140, except one second additional cycle amidst the Maunder
minimum (1650) which, giving its low quality factor according to Usoskin et
al. (2021), should also be taken with a grain of salt.

While we do not go so far as to claim perfect evidence for a clocked solar
dynamo, we strongly refute the claim of Weisshaar, Cameron and Schüssler
(2023) for a high statistical significance of a non-clocked process. By
contrast, the likely one-to-one match of the 14C data with Schove’s series
would imply the latter to be (more or less) correct which, in turn, would also
impugn any criticism of Schove’s “9 per century” rule. Moreover, it remains to
be seen how the arguments of Nataf (2022) and Weisshaar, Cameron and Schüssler
(2023) against clocking could be reconciled with the algae data of Vos et al.
(2004).

At any rate, in this article we consider the phase-stability of the Schwabe
cycle as a serious working hypothesis, for which it seems worthwhile to find a
reasonable physical explanation. If, someday in the future, new data show up
to give unambiguous evidence to the contrary, we will be the first to declare
this paper, as well as its predecessors (Stefani et al., 2016, 2018; Stefani,
Giesecke and Weier, 2019; Stefani et al., 2020a; Stefani, Stepanov, and Weier,
2021; Klevs, Stefani, and Jouve, 2023), as misleading and futile.

##  3 Rieger

We start with the spectrum of the tidal forces, computed from the NASA/NAIF
ephemerides DE430 (Folkner et al., 2014). Evidently, the most prominent peaks
in Figure 1 correspond to the two-planet spring tides of Venus-Jupiter (118.50
days), Earth-Jupiter (199.44 days), and Venus-Earth (291.96 days), see
Scafetta (2022). A few more peaks at yet smaller periods are due to the spring
tides of Mercury with Jupiter (44.9 days), Earth (58 days) and Venus (72.3
days).

Figure 1.: PSD of the tidal forces exerted by all eight planets and Pluto. The
main peaks are formed by various two-planet spring-tides except that at 11.86
years which stems from Jupiter’s orbit, and one at 983 years which results in
a complicated manner from the joint action of Venus, Earth, Jupiter and
Saturn.Fig:fig1

The next prominent peak, at 9.93 years, corresponds to the Jupiter-Saturn
spring tide which follows from the rosette-shaped, 19.86-year periodic motion
of the Sun around the barycenter of the solar system (which, in turn, forms
the most dominant peak in the PSD of the angular momentum, see Figure 3c in
Stefani et al. (2020a)). Note also the strong 11.86-year peak corresponding to
the orbit of Jupiter. As noticed by several authors (Nataf, 2022; Cionco,
Kudryavtsev, and Soon, 2023), at this level of linear tidal action there is no
significant peak showing up at 11.07 years, a point we will come back to later
on. For an explanation of the last peak at 983 years, resulting in a
complicated manner from the joint action of Venus, Earth, Jupiter and Saturn,
see Scafetta (2022).

Remarkably, the three dominant peaks at periods corresponding to the Venus-
Jupiter, Earth-Jupiter and Venus-Earth spring tides belong to the band of
Rieger-type periods. After the initial identification of a 154-day periodicity
in the occurrence of hard solar flares by Rieger et al. (1984), those
periodicities have received enormous attention (Bai and Sturrock, 1991;
Ballester, Oliver, and Baudin, 1999; Dzhalilov et al., 2002; Bilenko, 2020;
Korsós et al., 2023).

In this context, it had soon be realized that Rieger-type periods correspond
to the typical periods of magneto-Rossby waves during the solar maximum
(Zaqarashvili, 2010), a topic which has been pursued intensely ever since
(Gurgenashvili et al., 2016; Marquez-Artavia, Jones, and Tobias, 2017; Dikpati
et al., 2018; Gachechiladze et al., 2019; Dikpati et al., 2020, 2021a, 2021b;
Gurgenashvili et al., 2021). Elaborating on this idea (and also with a side
view on a similar spring-tide excitation of Rossby-wave in the Earth’s
atmosphere, as discussed recently by Best and Madrigali (2015)), in this
section we use the methods of Horstmann et al. (2023) to ask specifically what
amplitudes of magneto-Rossby waves could be generated by the individual two-
planet spring tides of Venus-Jupiter, Earth-Jupiter, and Venus-Earth. Further
below, we will consider the nonlinear effects resulting from the combination
of those individual terms.

Anticipating the typical Rieger-type periods involved, we focus here on the
most relevant non-equatorial magneto-Rossby waves. At this point, a careful
distinction must be made between two different types of magneto-Rossby waves.
On the one hand, there are slow magneto-Rossby waves, which can reach
Gleissberg periodicities (Zaqarashvili, 2018) and differ fundamentally from
classical Rossby waves in that they propagate progradely instead of
retrogradely along the latitudes. On the other hand, also retrograde magneto-
Rossby waves exist on the classic dispersion branch, which are more similar to
hydrodynamic Rossby waves but differ notably in their natural frequencies
being right in the range of Rieger periodicities. In the following, we will
focus exactly on the latter fast magneto-Rossby waves, which are governed in
the framework of the Cartesian $\beta$-plane approximation (with longitudinal
coordinate $x$ and latitudinal coordinate $y$) by the following forced wave
equation (Horstmann et al., 2023) for the latitudinal velocity component
$v:=v_{\rm y}=v_{\phi}$:

$\displaystyle\square^{2}_{v_{A}}v-C_{0}^{2}\square_{v_{A}}\Delta
v+f_{0}^{2}\frac{\partial^{2}v}{\partial
t^{2}}-C_{0}^{2}\beta\frac{\partial}{\partial x}\frac{\partial v}{\partial
t}+2\lambda\frac{\partial}{\partial t}\square_{v_{A}}v-\lambda
C_{0}^{2}\Delta\frac{\partial v}{\partial
t}+\lambda^{2}\frac{\partial^{2}v}{\partial t^{2}}$
$\displaystyle=f_{0}\frac{\partial}{\partial x}\frac{\partial^{2}V}{\partial
t^{2}}-\lambda\frac{\partial}{\partial y}\frac{\partial^{2}V}{\partial
t^{2}}-\frac{\partial}{\partial t}\frac{\partial}{\partial
y}\square_{v_{A}}V\;.$ (1)

Here we use the definitions $C_{0}=\sqrt{gH_{0}}$ and
$v_{A}=B_{0}/\sqrt{4\pi\rho}$ for the gravity wave and Alfvén velocities,
respectively (with $g$, $H_{0}$, $B_{0}$ and $\rho$ denoting the reduced
gravity, height of the tachocline layer, toroidal magnetic field and density
of the plasma, respectively), and
$\square_{v_{A}}:=\partial_{t}^{2}-v_{A}^{2}\partial_{x}^{2}$ for the
d’Alembert operator with respect to Alfvén waves. Further,
$f_{0}=2\Omega_{0}\sin(\phi_{0})$ and
$\beta=(2\Omega_{0}/R_{0})\cos(\phi_{0})$ specify the Coriolis and Rossby
parameters with $\Omega_{0}$ and $R_{0}$ denoting the Sun’s angular velocity
and the mean radius of the tachocline. Finally, $\lambda$ is an empirical
damping parameter and $V$ denotes the external tidal potential. Since the
projections of spring-tide envelope potentials onto the $\beta$-plane are
quite delicate, we use a simplified potential of a single tide-generating body
projected on the $\beta$-plane at mid-latitudes $\phi_{0}=45^{\circ}$

$\displaystyle V$ $\displaystyle=$ $\displaystyle
K\left(\frac{1}{2}+\frac{y}{R_{0}}\right)\left[1+\cos\left(\frac{2x}{R_{0}}+\Omega_{\rm
st}t\right)\right],$ (2)

which oscillates here with the effective spring-tide frequency $\Omega_{\rm
st}$ instead of the single planet’s orbit frequency. The tidal forcing
amplitude is denoted by $K$, for which we use only the value of Jupiter
$K=504\,{\rm cm}\,{\rm s}^{-2}$ for the sake of comparability with later
result. This approach allows us to carry out conservative order-of-magnitude
estimates of Rossby wave amplitudes since the tidal peaks associated with the
discussed spring-tide frequencies have all similar or even higher powers than
Jupiter’s peak, see Figure 1. A more accurate computation of wave excitation
by spring tides has to be left for future work.

The response of magneto-Rossby waves to tidal forcing depends essentially on
the damping factor $\lambda$ that is mainly (but not exclusionary) related to
the (turbulent) viscosity of the fluid, which in turn is a widely unknown
quantity. In order to get a first clue on the possible reactions, in Figure 2
we consider five different damping factors, the higher four ones corresponding
to $\lambda/\Omega_{\rm st}=10^{-1}...10^{-4}$ and refer to hypothetical
lifetimes of freely decaying waves, the lowest one (here: $4\cdot 10^{-6}$)
corresponding to an estimation of boundary layer damping
$\lambda\sim\sqrt{\nu\Omega_{\rm st}}/H_{0}$, which is orders of magnitudes
larger than internal damping $\lambda\sim\nu/R_{0}^{2}$ in shallow wave
systems and often the dominating source of viscous dissipation. We have
applied the molecular viscosity of $\nu=10^{2}$ cm2/s, which is, on the one
hand, still conservative in comparison with the even smaller estimate 27 cm2/s
of Gough (2007). On the other hand, in strongly turbulent flows much larger
eddy viscosities must be adopted depending on the underlying turbulent scales,
which are widely speculative for tachoclinic waves. For that reason this
estimation delineates an upper limit for resonant wave excitation. In this
example, we consider the 118-day period of the Venus-Jupiter spring tide, but
use, as said above, only the amplitude of Jupiter’s tidal forcing. In doing
so, we follow two solar cycles during which the toroidal field (entering the
Alfvén speed $v_{A}$) is assumed to have a sinusoidal dependence with an
amplitude of 40 kG. Then, at each magnetic field strength, we search for that
gravity-wave velocity $C_{0}$ at which the wave response becomes maximum
(Figure 2b). This velocity corresponds to a certain height within the
tachocline (with higher velocities corresponding to lower heights and vice
versa). While, quite generally, the amplitude of the magneto-Rossby wave grows
with the magnetic-field strength, we observe a strong dependence on the
damping parameter $\lambda$. For the lowest three damping rates, the triggered
velocity can reach values between a moderate 10 cm/s and a whopping 20 m/s
which is indeed remarkable.

For the moment we ignore the principle possibility that the damping factor
might still increase with the magnetic field strength, which would play an
additional role in the resonance effects to be discussed later.

Figure 2.: (a) Maximum velocity of the magneto-Rossby wave, induced by a
tidal-force amplitude of Jupiter, but for an assumed forcing period of 118
days (Venus-Jupiter spring-tide), in dependence on an artificial 22-year cycle
of the toroidal magnetic field. (b) For each field strength, there is a
different gravity-wave velocity, corresponding to a certain depth within the
tachocline, which leads to the optimum reaction according to (a). The response
is computed for five different damping parameters, the smallest one
corresponding to a viscosity of 100 cm2/s.Fig:fig2

Figure 3 shows the respective outcomes for the forcing period of 199 days of
the Earth-Jupiter spring-tide. Evidently the results are not much different
from those of Figure 2, except that the reached velocities are now a bit
higher111Interestingly, this enhancement of the wave response with the period
would be roughly compensated by the slightly lower spring-tides of the Earth-
Jupiter system.

Figure 3.: Same as Figure 2, but for an assumed forcing period of 199 days
(Earth-Jupiter spring tide).Fig:fig3

Things are changing, however, when we go over to the 292-day spring-tide
period of Venus-Earth (Figure 4). While the excited wave velocities (reaching
up to 100 m/s) are generally still higher than in Figure 3, at the highest
magnetic fields we observe now a breakdown of the wave excitation. Evidently,
at this forcing period we cross the maximum field strength beyond which there
is no layer of the tachocline where the waves can be excited (at least for the
parameters used in our model). Further below, when treating the
synchronization problem of the dynamo, this behaviour will play a role in
justifying the field-dependent resonance term for the periodic part of the
$\alpha$-effect.

Figure 4.: Same as Figure 2, but for an assumed forcing period of 292 days
(Venus-Earth spring-tide). Note the breakdown of the wave excitation for the
strongest magnetic fields.Fig:fig4

At this point, we add another Figure 5 related to the tidal effects of
Mercury. It had been noticed previously (Hung, 2007) that the tidal action of
Mercury is not much weaker than that of Earth, making its omission in
synchronization models always a bit suspicious. As shown now in Figure 5, at
the Mercury-Venus spring tide of 72 days (which leads to relatively modest
wave amplitudes anyway) we get a similar dying-out effect as observed for the
case of 292 days, but now for the weakest magnetic fields. In the extreme case
of the Mercury-Jupiter spring-tide period of 45 days, we have checked that
there is no wave excitation at all (not shown here). Obviously, for both too
long and too short tidal forcings we face the situation that the Sun can
simply not “understand” what the planets are “telling” it since its resonance-
ground - in the form of retrograde magneto-Rossby waves - does not function
anymore.

Figure 5.: Same as Figure 2, but for an assumed forcing period of 72 days
(Mercury-Venus spring tide). Note the breakdown of the wave excitation for the
weakest magnetic fields.Fig:fig5

What we have learned so far is that the three two-planet spring-tide
periodicities of the tidally dominant planets Venus, Earth and Jupiter fit
amazingly well to the natural periods of magneto-Rossby waves in the
tachocline. On the very high-frequency side (applying to Mercury-Jupiter)
waves are not being excited at all, while on the low-frequency side (Venus-
Earth) the waves were shown to die out when the magnetic field becomes too
strong. Moreover, the weakening of the spring-tides of Earth-Jupiter (199
days) and Earth-Venus (292 days) compared to that of Venus-Jupiter (118 days)
is a sort of compensated by the increasing reaction of the triggered magneto-
Rossby wave at those longer periods. This effect will become of relevance
further below when we discuss potential non-linear effects of the three
superposed waves.

For the moment, however, it is tempting to ask whether the identified spring-
tide periods do actually show up in the solar data. While, on the first
glance, the very 154-day period that was once found by Rieger et al. (1984)
does not fit to any of the tidal periods just discussed, other data look more
promising. In his spherical harmonic decomposition of the solar magnetic
field, Knaack (2005) had identified quasi-periodicities grouped around 300-320
days, 220-240 days, 170 days, and 100-130 days. In their analysis of the total
solar irradiance (TSI) during cycle 23, Gurgenashvili et al. (2021) had seen
two significant maxima around 115 days and 180 days. Most interestingly, in
cycle 24 a significant Rieger peak close to 150 days showing up in the VIRGO
and SATIRE-S TSI data was accompanied by a strong and clear peak at 195 days
in the sunspot area data (their Figure 2). A similar distinction appeared when
comparing the sunspot areas in the Northern and Southern hemisphere (their
Figure 3). With view on such ambiguities, it is not completely clear whether
the attribution of 185-195-day periods to weak cycles and of 155-165-day
periods to strong cycles (Gurgenashvili et al., 2016) always applies.

One should also keep in mind that the link between the intrinsic periods of
the (tidally triggered) waves and the observable periods might be a bit
indirect. One of the intervening factors to consider is the rise time of flux
tubes from the tachocline to the solar surface whose field-dependence is
highly nontrivial, and, very likely, non-monotonic (see Figure 8 of Weber et
al. (2011)). With only a very few Rieger-type periods covering the short
strong-field interval of a cycle, such a field-dependence of the rise times
could easily smear out any sharp periods of the underlying waves on the
tachocline level (see, e.g., Figure 1 of Gurgenashvili et al. (2016)). In this
respect it is interesting to note that the typical Rieger period of some 155
days, say, lies quite in the middle between the 118 day of the Venus-Jupiter
spring tide the 199 days of the Earth-Jupiter spring-tide, so that this signal
might be fed by a rise-time related modification of both periods.

Apart from the very temporal period, the azimuthal dependence of the observed
quantities represents another clue when it comes to the identification of
magneto-Rossby waves. While in our model the latter are supposed to have the
$m=2$ azimuthal dependence as the tidal forcing, the corresponding dependence
for the observed data is less clear, despite the fact that active longitudes
should be related to the dominant wavenumbers of the tachocline bulges that
contain toroidal fields (Dikpati et al., 2018). In many data (Gyenge et al.,
2016, 2017; Dikpati et al., 2018), one can find both $m=1$ and $m=2$
contributions, typically with some dominance of the $m=1$ part (see also
Raphaldini et al. (2023)). The latter, however should not be considered as an
exclusionary argument against a dominant $m=2$ wave at the tachocline level,
since threshold effects like the launching of flux tubes are certainly
sensitive to slight symmetry breakings. Most interesting results on the
azimuthal dependence have been obtained by Bilenko (2020): distinguishing
between zonal ($m=0$) and sectorial harmonics ($m=n$), this author had found,
in cycle 22, a dominant $m=2$ signal with a period of 170-190 days, and in
cycle 23 a corresponding $m=2$ structure in the range of 160-200 days.

While, therefore, neither the temporal nor the azimuthal dependence of the
observations give perfect evidence for tidally-triggered magneto-Rossby waves,
they both entail sufficient indications that may justify further data analyses
to check this hypothesis.

Another feature to be shortly mentioned here is the appearance of longer
periods such as 1.3 years (Howe et al., 2000) or 1.7 years (Korsós et al.,
2023) which both are definitely beyond the range of excitability of retrograde
magneto-Rossby waves as discussed above. In this respect it seems worthwhile
to consider the possibility of intervening non-linearities, e.g., tachocline
nonlinear oscillations (Dikpati et al., 2018, 2021a).

Two types of such nonlinear wave couplings were recently studied in detail by
Raphaldini et al. (2019, 2022). In either case they involve quadratic
interactions between the waves which might lead to various frequency and
wavenumber resonances. With respect to the latter, we point to the analysis of
Bilenko (2020) who showed the occurrence of dominant $m=4$ modes at the solar
maximum, connected with strong equatorially symmetric $m=0$ modes (Bilenko’s
Figure 4d,e). Obviously, such a combination of $m=4$ and $m=0$ modes is
exactly what would be expected from a quadratic interaction of the tidally
triggered $m=2$ modes as considered so far.

While we refrain here from a detailed computation of the full spatio-temporal
structure of those nonlinear terms (e.g., in equation systems (25-27) or
(28-32) of Raphaldini et al. (2019)), we restrict our attention to the
temporal dependence of the square of the sum of three sine-functions,
representing different magneto-Rossby waves,

$\displaystyle s(t)=\left[\cos\left(2\pi\cdot\frac{t-t_{\rm VJ}}{0.5\cdot
P_{\rm VJ}}\right)+\cos\left(2\pi\cdot\frac{t-t_{\rm EJ}}{0.5\cdot P_{\rm
EJ}}\right)+\cos\left(2\pi\cdot\frac{t-t_{\rm VE}}{0.5\cdot P_{\rm
VE}}\right)\right]^{2}$ (3)

with the two-planet synodic periods $P_{\rm VJ}=0.64884$ years, $P_{\rm
EJ}=1.09207$ years, $P_{\rm VE}=1.59876$ years, and the epochs of the
corresponding conjunctions $t_{\rm VJ}=2002.34$, $t_{\rm EJ}=2003.09$, and
$t_{\rm VE}=2002.83$ being adopted from Scafetta (2022).

What we then see in Figure 6a is the appearance of a beat period of around 1.6
years. If we were to skip the Venus-Earth spring tide in Equation (3) (e.g.,
for those times of its disappearance when the magnetic field is too strong,
see Figure 4), we would obtain Figure 6b showing a beat period of 291 days
which, in turn, corresponds to the (omitted) Venus-Earth spring-tide222This
correspondence can be understood by expressing the two-planet spring tide
periods $P_{\rm ij}$ by the individual periods $P_{\rm i}$ and $P_{\rm j}$..
It remains to be seen whether these periods might be related to the periods of
0.8-0.9 years and 1.7-1.8 years as observed by Korsós et al. (2023).

Figure 6.: (a) Amplitude of the squared sum of the three individual spring-
tide contributions of Venus-Jupiter, Earth-Jupiter, and Venus-Earth. The
dominating spiky behaviour corresponds to a period of appr. 1.6 years. (b) The
same, but with the contribution from the Venus-Earth spring tide being
omitted. The dominating spiky behaviour corresponds here to the beat period of
292 days.Fig:fig6

##  4 Schwabe

Having seen that the spring tides of two planets are potentially capable of
triggering magneto-Rossby waves with amplitudes of the order of m/s or even
more, we will now try to figure out how this wave energy, once “harvested”,
can be utilized by the solar dynamo for synchronizing its Schwabe cycle. The
question of whether this cycle is synchronized by the planets or not has a
long history, going back to Wolf (1859) and de la Rue et al. (1872), and being
further pursued by Bollinger (1952), Takahashi (1968), Wood (1972), Öpik
(1972), Okal and Anderson (1975), Condon and Schmidt (1975), Dicke (1978),
Hoyng (1996), Zaqarashvili (1997), Palus et al. (2000), De Jager and Versteegh
(2005), and Callebaut, de Jager, and Duhau (2012). However, the amazingly
precise correspondence with the three-planet spring tides of 11.07 years was
discussed only recently by Hung (2007), Scafetta (2012), Wilson (2013) and
Okhlopkov (2016).

Apart from some more exotic models relying on changed rates of nuclear fusion
in the Sun’s core (Wolff and Patrone, 2010; Scafetta, 2012), there are
basically four mechanisms which seem suitable to provide a discernible effect
on the dynamo. The first one assumes that the ellipticity of the Sun’s
barycentric motion leads to periodic changes of its differential rotation
(Zaqarashvili, 1997). The second one relies on the high sensitivity of the
field storage capacity within the weakly sub-adiabatic tachocline, which might
easily react even to weak (pressure) disturbances by releasing more or less
magnetic flux tubes for its rise to the Sun’s surface. This concept,
originally introduced by Abreu et al. (2012) (based on Ferriz Mas, Schmitt,
and Schüssler (1994), was recently tested in a 2D Babcock-Leighton variant of
a synchronized dynamo model by Charbonneau (2022) (relying on the time-delay
concept of Wilmot-Smith et al. (2006)). Exhibiting the same type of parametric
resonance as previously observed by Stefani, Giesecke and Weier (2019), this
promising result superseded the skeptical judgment of the synchronization
suitability of Babcock-Leighton type models that was based on an evidently too
simple 1D model of Stefani et al. (2018).

A third potential mechanism for dynamo synchronization had emerged with the
observation that the kink-type, current-driven Tayler instability (TI)
(Tayler, 1973; Seilmayer et al., 2012) is prone to intrinsic helicity
oscillations (Weber et al., 2013, 2015), at least for small magnetic Prandtl
numbers (which applies to the tachocline). While first identified in the
simplified setting of TI in a non-rotating, full cylinder, a similar
oscillatory behaviour was recently shown to also appear in a much more
realistic 3D-model of the tachocline (Figure 16 in Monteiro et al. (2008)). A
key observation of Stefani et al. (2016) was then that those helicity
oscillations of the TI (with its azimuthal wave number $m=1$) are sensitive to
entrainment by tide-like ($m=2$) perturbations. For the not dissimilar case of
an $m=1$ Large Scale Circulation in a Rayleigh-Bénard convection problem this
concept of helicity synchronization by tide-like forcing was recently
exemplified in a liquid metal experiment (Jüstel et al., 2020, 2022).

Slightly modifying the concept of directly forced helicity synchronization, we
focus here on the fourth possibility that the (tidally triggered) magneto-
Rossby waves themselves give rise to an $\alpha$-effect, a possibility that
was first studied by Avalos-Zuñiga, Plunian and Rädler (2009). While the
$\alpha$-effect for stationary Rossby waves is typically zero (due to the 90∘
shift of vertical velocity and vorticity) these authors found a non-vanishing
$\alpha$-effects for the case of drifting waves. This is indeed the scenario
we are dealing with in this paper, and it remains to be seen how much of the
$\alpha$-effect is entailed in the superposition of the three magneto-Rossby
waves as excited by the two-planet spring tides discussed above.

Turning this question upside-down, in Klevs, Stefani, and Jouve (2023) it was
asked how much of the synchronized $\alpha$-effect would actually be needed to
entrain the entire solar dynamo by parametric resonance. Based on a rather
conventional 2D $\alpha-\Omega$ dynamo model, including meridional
circulation, this value was shown to lay in the range of some dm/s or even
less (with the resistivity jump between tachocline and convection zone being
the most decisive, yet widely unknown factor).

This consistency between the scale of m/s which results for the velocity of
tidally excited magneto-Rossby waves and the scale of dm/s that would be
required for a dynamo-synchronizing $\alpha$-effect is indeed encouraging.
Admittedly, it is still a long way to corroborate this link in more detail.
For a first glance on this issue, we will again restrict ourselves to some
simple considerations, leaving a detailed computation of the helicity, and the
$\alpha$-effect connected with it (Avalos-Zuñiga, Plunian and Rädler, 2009),
to future work.

The central question in this respect is how the 11.07-year period, which is
known to have no significant peak in the total tidal potential (see Figure 1
and Okal and Anderson (1975), Nataf (2022), Cionco, Kudryavtsev, and Soon
(2023)) might still emerge from the interactions of the much shorter two-
planet spring-tides. The classical answer to that question is given in
Scafetta’s formula for the period $P_{\rm VEJ}$ of the three-body spring tide
(Scafetta, 2022)

$\displaystyle P_{\rm VEJ}=\frac{1}{2}\left[\frac{3}{P_{\rm
V}}-\frac{5}{P_{\rm E}}+\frac{2}{P_{\rm J}}\right]^{-1}$ (4)

where $P_{\rm V}=224.701$ days, $P_{\rm E}=365.256$ days, and $P_{\rm
J}=4332.589$ days are the sidereal orbital periods of Venus, Earth and
Jupiter, respectively. The vector $(3,-5,2)$, appearing in Equation (4), can
also be written as $(3,-5,2)=3(1,-1,0)-2(0,1-1)$, indicating the frequency of
the beat being created by the third harmonic of the synodic cycle between
Venus and Earth and the second harmonic of the synodic cycle between Earth and
Jupiter (Scafetta, 2022).

Figure 7.: Signal of various combinations of spring tides. (a) Scafetta’s
equation (5) showing the sum of the third harmonic of the synodic cycle
between Venus and Earth and the second harmonic of the synodic cycle between
Earth and Jupiter. (b) Squared sum of the individual (and equally weighted)
spring-tide contributions of Venus-Jupiter, Earth-Jupiter, and Venus-Earth
according to Equation (3). (c) Same, but without the Venus-Earth contribution
(i.e., last term in Equation (3)). In either case the three-planet spring-tide
period of 11.07 years becomes visible.Fig:fig7

The emergence of this beat is illustrated in Figure 7a in which we replot the
same function

$\displaystyle s(t)=\cos\left(2\pi\cdot 2\cdot\frac{t-t_{\rm EJ}}{0.5\cdot
P_{\rm EJ}}\right)+\cos\left(2\pi\cdot 3\cdot\frac{t-t_{\rm VE}}{0.5\cdot
P_{\rm VE}}\right)$ (5)

as in Figure 2A of Scafetta (2022), showing a clear-cut beat period of 11.07
years.

Convincing as this curve might look, it is not entirely clear how exactly this
very particular combinations of third and second harmonics of the two-planet
spring tides might be of any physical relevance. An important aspect was
attributed by Scafetta to the feature of “orbital invariance”, meaning that
the identity $3-5+2=0$ involves that all parts of the differentially rotating
body feel the same forcing.

Interestingly, the very same 11.07-year beat period shows up when we only
consider the square of the sum of the three cosine functions of Equation (3)
which is indeed expected to occur, e.g., in typical quadratic functionals of
the velocity perturbations, such as, e.g., the Eliassen-Palm flux that is
well-known in meteorology (Trenberth, 1986), pressure perturbations, or the
helicity. This is clearly seen in Figure 7b which illustrates again the signal
from Equation (3), this time, however, for the enlarged interval of 200
years333It would be an interesting, if tedious, exercise in applying the
addition theorems for trigonometric functions to quantify how much of the
“ideal” expression (5) is contained in the square of three functions according
to Equation (3).. And even in case that the Venus-Earth 291-days spring tide
is neglected (e.g., due to a too strong magnetic field) it appears that we
could obtain the same 11.07 years beat, as evidenced in Figure 7c.

However, the latter conclusion turns out to be a bit pre-mature. With view on
synchronized dynamo models such as in Klevs, Stefani, and Jouve (2023), we
should presumably focus on the axi-symmetric component of the quadratic term
in Equation (3), very likely with some phase shift of one of the two factors
(which would be needed when computing the helicity). For the simplest case
without phase shift between the two factors, we show in Figure 8a the
azimuthal average

$\displaystyle S(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\left[\cos\left(2\pi\cdot\frac{t-t_{\rm
VJ}}{0.5\cdot P_{\rm VJ}}+2\varphi\right)+\cos\left(2\pi\cdot\frac{t-t_{\rm
EJ}}{0.5\cdot P_{\rm EJ}}+2\varphi\right)\right.$ (6)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left.\cos\left(2\pi\cdot\frac{t-t_{\rm
VE}}{0.5\cdot P_{\rm VE}}+2\varphi\right)\right]^{2}$

wherein the phase $2\varphi$ reflects the $m=2$-character of the tidally
triggered waves. As we can see, the 11-07-yr period is still visible in this
integral, just as it was in Figure 7b. Yet, things are changing when we skip
in Equation (6) the last term in the sum, corresponding to the Venus-Earth
spring tide. In this case, the 11.07-year period disappears completely, so
that Figure 8b looks very different from Figure 7c. In the Appendix, we will
explain these different behaviours in terms of the azimuthal dependencies of
the square under the integral in Equation (6), and relate them to Scafetta’s
notion of “orbital invariance” which refers to an action that is “being
simultaneously and coherently seen by any region of a differentially rotating
system” (Scafetta, 2022).

Figure 8.:  (a) Azimuthally averaged squared sum of the individual spring-tide
contributions of Venus-Jupiter, and Earth-Jupiter and Venus-Earth according to
Equation (6). (b) Same, without the Venus-Earth contribution (i.e., last term
in the sum of Equation (6)). The three-planet spring-tide period of 11.07
years remains visible only in case (a)Fig:fig8

We refrain here from corroborating more details on how the superposition of
three (or two) tidally triggered magneto-Rossby waves would generate helicity,
or an $\alpha$-effect, for that matter. In the very simplest case, the
azimuthally averaged helicity would integrate to zero, when the second factor
under the integral in Equation (6) is endowed with an additional phase shift
of 90∘ that comes into play when taking the curl of the velocity (though a bit
trivial, this vanishing is also illustrated in the Appendix). However, as
shown by Avalos-Zuñiga, Plunian and Rädler (2009), for more complicated
geometries some value of $\alpha$ still remains, and there are good reasons to
expect it to have a similar 11.07 year modulation as that in Figure 8a.

At any rate: with view on the parametric resonance of the underlying
conventional dynamo having a natural period not too far from 22 years, the
dynamo will react preferably on perturbations with periods around 11 years
(see Figure 10 in Stefani, Giesecke and Weier (2019), Figure 10 of Charbonneau
(2022), and Figure 6 in Klevs, Stefani, and Jouve (2023)). For this resonance,
azimuthally averaged quantities such as in Figure 8a will presumably play the
dominant role.

A this point, the discerning reader will have noticed a slight, but important
conceptual shift. In all previous works (Stefani et al., 2016, 2018; Stefani,
Giesecke and Weier, 2019; Stefani, Stepanov, and Weier, 2021; Klevs, Stefani,
and Jouve, 2023), the 11.07-year periodicity was implemented as a resonance
term with the generic field-dependence $\sim B^{2}/(1+B^{4})$ that was
interpreted in terms of an optimal reaction, on the Schwabe time-scale, of the
TI to the amplitude of the field. Now we re-interpret the very same resonance
term as an optimal response of the magneto-Rossby waves to the magnetic field
on the Rieger-type time-scale. As was shown above in Figure 4, not only too
low fields (as in Figure 5) prevent the wave from being excited, but too high
fields do so as well. This is all the more plausible as magnetic fields might
also contribute to the damping factor $\lambda$, resulting in another
reduction of the wave amplitudes with increasing field.

This new interpretation opens up the possibility that very weak magnetic
fields could possibly prevent the excitation of Rieger-type oscillations,
which might even lead to a loss of synchronization. In this respect it would
be interesting if the hypothesized additional cycle around 1650 (Usoskin et
al., 2021), i.e. deep in the Maunder minimum, could be confirmed by
independent data. A closely related question refers to the observation of
Reinhold et al. (2020) that the Sun’s activity appears to show a significantly
lower variability than other stars with near-solar temperatures and rotation
periods. In this regards, and given some interesting applications in other
fields (Pontin et al., 2014), it is not unlikely that this lower variability,
or noise level, indeed results from parametric resonance.

##  5 Suess-de Vries

There is some tradition of linking long-term temporal variabilities of the
solar cycle to correspondingly long periods of planetary influences (Jose,
1965; Charvatova, 1997; Landscheidt, 1999). One of the most detailed attempts
in this direction was that of Abreu et al. (2012) who had identified quite a
couple of common periodicities. Yet, their approach was soon criticized, for
various reasons, by Cameron and Schüssler (2013) and Poluianov and Usoskin
(2014). More recently still, Charbonneau (2022) put into question the entire
approach of linking long-term cycles with correspondingly long periodicities
of the planetary system by arguing that “(b)ecause the low frequency response
is internal to the dynamo itself, it is similar for incoherent stochastic
forcing and coherent external periodic forcing”. This lead him to the
conclusions “that if external forcing of the solar dynamo of any origin is
responsible for the centennial and millennial modulations of the magnetic
activity cycle, the required forcing amplitudes should be well beyond the
homeopathic regime in order to produce a detectable signature in the presence
of other sources of modulation, whether deterministic or stochastic.”

Figure 9.: Comparison between the spectra of climate-related observations and
numerical simulations. (a) Spectrum of varved lacustrine-sediment data from
Lake Lisan, adapted from Figure 3A of Prasad et al. (2004). The dashed black
line indicates the 95 % confidence interval, the significant periodicities are
labeled in years. The ordinate axis [a.u.] is linearly scaled. (b) Spectra of
a 1D $\alpha-\Omega$-dynamo model (here, the ordinate axis is logarithmic).
The green curve (only Jupiter and Saturn included in the computation of the
orbital angular momentum) and the blue curve (all planets included) correspond
to those in Figure 9 of Stefani, Stepanov, and Weier (2021) (with periods just
replaced by frequencies). The red curve is similar to the blue one, but
incorporates some noise in the $\alpha$-term. For more details, in particular
regarding the origin of the indicated periodicities, see the main text. Note
the good agreement of the main peaks for the observations (a) with those of
the numerical predictions (b), in particular regarding the sharp Suess-de
Vries cycle at 193 years.Fig:fig9

But what if the external forcing acts on much shorter than centennial and
millennial periodicities, and the latter show up only as beat periods of the
former? This concept, which traces back to ideas of Solheim (2013) and Wilson
(2013), was pursued in Stefani et al. (2020a) and Stefani, Stepanov, and Weier
(2021), where a combination of a primary 11.07-year forcing of the
$\alpha$-effect with the 19.86-year period related to the barycentric motion
of the Sun led to the emergence of an 193-year period. In view of very similar
periods identified by Prasad et al. (2004), Richards, Rogers and Richards
(2009), Lüdecke, Weiss and Hempelmann (2015), and Ma and Vaquero (2020) this
might indeed be the Suess-de Vries cycle, although its period is often quoted
with the slightly larger value of 200 or even 208 years. More complicated
still, both in terms of observation and modelling, is the case of the
Gleissberg cycle(s) whose occurrence was, in Figure 9 of Stefani, Stepanov,
and Weier (2021), attributed partly to the second and third harmonics of the
193-year period, and partly to the additional effects of Saturn-Uranus and
Saturn-Neptune.

In Figure 9, we compare these results with the spectrum of climate-related
varved-sediment data from Lake Lisan, Dead Sea Rift, as obtained by Prasad et
al. (2004) for the time interval 26.2-17.7 (calendar) ka (Figure 9a). The
green and blue curves in Figure 9b correspond to those in Figure 9 of Stefani,
Stepanov, and Weier (2021), except that the abscissa is changed from period to
frequency to ease the comparison with the sediment data. When using only the
positions of Jupiter and Saturn for computing the Sun’s orbital angular
momentum (green curve), we basically obtain the dominant $22.14\times
19.86/(22.14-19.86)=192.85\approx 193$ years period and its second and third
harmonics at $\approx 96$ and $\approx 64$ years, plus one minor peak at
$89.44\approx 89$ years which corresponds to the beat period of Saturn’s
orbital period (29.424 years) with the Hale period of 22.14 years. However,
when taking into account all planets (blue curve) we obtain, first of all,
three additional peaks at 82.7 years (beat period between the Jupiter-Neptune
synodic period of 12.78 years with 11.07 years), at 57.8 years (beat period of
the Saturn-Neptune synodic period of 35.87 years with 22.14 years), and at
55.8 years (beat period of the Jupiter-Uranus synodic period of 13.91 years
with 11.07 years). In addition to that, we observe the emergence of a third
subharmonic of 193 years at 579 years, and three fourths of 193 years at 144
years, both of which were absent for the pure Jupiter-Saturn forcing (at
present, we do not have a simple explanation for this somewhat counter-
intuitive behaviour). To this, we add a third (red) curve which is
distinguished from the blue one only by the presence of some weak noise (with
noise intensity $D=0.01$ according to the notation Stefani, Stepanov, and
Weier (2021)). Adjacent to this red curve, we summarize all the periods as
they were just discussed in connection with the green and blue curves.

When comparing the numerical results for the solar dynamo (Figure 9b) with the
climate-related observations (Figure 9a), the most striking feature is the
nearly perfect agreement of the dominant and rather sharp Suess-de Vries
period. The - still reasonable - agreement for the Gleissberg-type periods
around 90 years and slightly below 60 years becomes more complicated by the
existence of some close-by peaks (indicated by the dashed red lines) in the
numerical model. Interestingly enough, a decent correspondence between the
periodicities around 144 and 579 years can also be seen. Although we do not
claim perfect accordance between our model and the observations, we find the
match between the peaks, in particular the sharp Suess-de Vries period, quite
remarkable.

Figure 10.: Appearance of a beat period of 200 years from a basic Hale cycle
of 22 years and an 19.82-year period related to the barycentric motion of the
Sun. (a) Three magnetic field components at $r=0.7$ during a 2500-year
interval. (b) Details of (a) for a 500-year interval. (c) PSD over the period
$T$, with a clear dominance of the 22-year cycle, a subdominant 200-year
cycle, and a Gleissberg-type cycle with 66.7 years. Note that the actual solar
values are 22.14-years, 19.86 years, and 193-years.Fig:fig10

In the following, we go beyond this simple 1D model and try to confirm the
emergence of the Suess-de Vries cycle in frame of the 2D $\alpha-\Omega$-model
as utilized by Klevs, Stefani, and Jouve (2023). In this model the primary
entrainment of the Schwabe cycle was accomplished by a periodic, and tidally
synchronized, contribution to the $\alpha$-effect in the tachocline region,
having the form of

$\displaystyle\alpha^{p}(r,\Theta,t)$ $\displaystyle=$ $\displaystyle
C^{p}_{\alpha}\frac{1}{\sqrt{2}}\sin^{2}\Theta\cos\Theta\left[1+{\rm
erf}\left(\frac{r-r_{c}}{d}\right)\right]\left[1-{\rm
erf}\left(\frac{r-r_{d}}{d}\right)\right]\times$ (7)
$\displaystyle\times\frac{2|{\bf{B}}(r,\Theta,t)|^{2}}{1+|{\bf{B}}(r,\Theta,t)|^{4}}\sin(2\pi
t/T_{f})\;,$

with $r_{c}=0.75$, $r_{d}=0.75$, $d=0.02$ (r is the dimensionless radius).
Here the resonance term has its maximum value equal to one at $|{\bf{B}}|=1$
which we re-interpret now as an optimum field strength for the excitation of
the three magneto-Rossby waves whose beat period forms the 11.07-year envelope
seen in Figure 8a.

In the 1D dynamo models of previous works (Stefani et al., 2020a; Stefani,
Stepanov, and Weier, 2021), the secondary 19.86-year period of the barycentric
motion was implemented as an additional $B^{3}$-term in the induction equation
representing a periodically changing field-storage capacity of the tachocline.
For the sake of simplicity, and in order to maintain the code structure of
Klevs, Stefani, and Jouve (2023), we decided to implement this effect as a
time-periodic change of the optimal excitation condition of the
$\alpha$-effect in Equation (7), with the underlying (if vague) assumption
that any change of the differential rotation (as discussed, e.g., by
Zaqarashvili (1997); Javaraiah (2003); Shirley (2006); Sharp (2013)) would
also modify the resonance condition of the waves (Gachechiladze et al.,
2019)). Specifically, we replace the simple resonance term in the second line
of Equation (7) by the more complicated one

$\displaystyle\frac{2(1+\epsilon\cos(2\pi
t/T_{B}))^{2}|{\bf{B}}(r,\Theta,t)|^{2}}{1+(1+\epsilon\cos(2\pi
t/T_{B}))^{4}|{\bf{B}}(r,\Theta,t)|^{4}}\sin(2\pi t/T_{f})$

which keeps the amplitude of the prefactor at 1 but oscillates the position of
this maximum between $1/(1+\epsilon)$ and $1/(1-\epsilon)$.

The results of the simulation for the particular case $\epsilon=0.9583$ are
shown in Figure 10. We stick here to the simplified Schwabe period
$T_{f}=11.00$ years, as in Klevs, Stefani, and Jouve (2023), and use
accordingly a slightly modified value $T_{B}=19.82$ years of the barycentric
period, in order to obtain also a simplified beat period of $11.00\times
19.82/(19.82-11)=200$ years.

We see that this beat period of 200 years indeed shows up in all field
components, measured here at $r=0.7$. We can identify, in particular, a clear
Gnevyshev-Ohl rule (Gnevyshev and Ohl, 1948) in terms if a succession of weak
and strong cycles, which is changing its order every 200 years, as also
observed for the solar cycle by Tlatov (2013)444Similar changes, on the Suess-
de Vries time-scale, of the North-South asymmetry were recently reported by
Mursula (2023)..

We should note, however, that for the particular model as chosen here such a
clear behaviour occurs only in a not too broad range of $\epsilon$, and that
more complicated behaviours are observed for other values. A more detailed
investigation of the corresponding parameter dependencies, as well as the
implementation of all planets in the angular momentum term, must be left for
future studies.

##  6 Bray-Hallstatt?

Usually, solar activity variations with time scales of 1-3 kyr are discussed
under the name of Eddy and Bray-Hallstatt cycles (Steinhilber et al., 2012;
Abreu et al., 2012; Soon et al., 2014; Scafetta et al., 2016; Usoskin et al.,
2016). However, it is by far not obvious whether the very concept of “cycles”
can be transferred at all from the decadal (Schwabe, Hale) and centennial
(Gleissberg, Suess-de Vries) to the millennial time scale555Note that the
underlying 14C and 10Be data bases have typical durations of only 10 kyr, or
just slightly longer, see Kudryavtsev and Dergachev (2020). In this context,
the analyses of certain ice drift proxies by Bond et al. (1997, 1999) had
provided evidence for millennial climate variability to occur in certain 1-3
kyr “cycles” of abrupt changes of the North Atlantic’s surface hydrography
which are closely related to corresponding grand minima of the solar dynamo
(Bond et al., 2001).

Figure 11.: (a) Signal according to Equation (7) in the interval -10000-10000.
(b) Orbital angular momentum of the Sun around the solar system’s barycenter
in the same interval, based on the DE431 ephemerides Folkner et al. (2014).
Note in either case the appearance of a 2318-yr period.Fig:fig11

Stefani, Stepanov, and Weier (2021) had shown a general tendency of the Suess-
de Vries cycle to undergo, for stronger forcings of the 19.86-year term,
abrupt breakdowns which resemble these grand-minima, or Bond events. Yet, such
chaotic events do not exclude some remaining regularity, perhaps driven by the
2318-year cycle of the Jupiter-Saturn-Uranus-Neptune system, according to
Equation (37) of Scafetta (2022),

$\displaystyle s(t)=\sin\left(2\pi\cdot\frac{t-t_{\rm JS}}{P_{\rm
JS}}\right)+\sin\left(2\pi\cdot\frac{t-t_{\rm SU}}{P_{\rm
SU}}\right)+\sin\left(2\pi\cdot\frac{t-t_{\rm SN}}{P_{\rm SN}}\right)$ (8)

with the two-planet synodic periods $P_{\rm JS}=19.8593$ years, $P_{\rm
SU}=45.3636$ years, $P_{\rm SN}=35.8697$ years, and the epochs of the
conjunctions $t_{\rm JS}=2000.48$, $t_{\rm SU}=1988.44$, and $t_{\rm
SN}=1989.54$.

This function is visualized in Figure 11a, which corresponds (apart from the
enlarged time interval) to Figure 5E in Scafetta (2022). Less formal than this
equation, though, is the actual orbital angular momentum, shown in Figure 11b,
which we have computed from the DE431 ephemerides described in Folkner et al.
(2014). Some obvious phase-shift notwithstanding, the same 2318-yr periodicity
shows up as in Figure 11a, which indeed speaks for the dominant role of the
two-planet synodic periods as indicated by Scafetta’s equation (7). It remains
to be seen in future work if the implementation of this real angular momentum
curve into the extended 2D $\alpha-\Omega$ dynamo model of Section 5 leads to
any noticeable signal on that time-scale, too.

Actually, such a 2300 year modulation had been found in proxies of the
galactic cosmic radiation by McCracken and Beer (2008). Interestingly, during
the observed cosmic radiation enhancements these authors also noticed a
strengthened Suess-de Vries cycle (their Figure 3), which is somehow
consistent with a similar numerical result in Figures 11 and 12 of Stefani,
Stepanov, and Weier (2021). This entire behavior is reminiscent of the concept
of supermodulation as described by Weiss and Tobias (2016).

As a side remark, note that a similar interplay of chaos and regularity had
been discussed as a stochastic resonance phenomenon in connection with the
reversal statistics of the geodynamo for which the 95-kyr Milankovic cycle of
Earth’s orbit excentricity seems to play a decisive role (Consolini and De
Michelis, 2003; Stefani et al., 2006; Fischer et al., 2008).

##  7 Summary and conclusions

In this paper, we have pursued the programme of explaining the various
temporal variabilities of solar activity as being triggered by gravitational
influences of the orbiting planets. In doing so, we focused not only on what
the Sun is being “told” by the the planets, but also on what it is able to
“understand”. On each timescale involved we have therefore asked about the
relevant intrinsic processes within the Sun that might be able to resonate
with the respective external forcings.

Starting from the shortest, Rieger-type timescales of some 100-300 days, we
have shown that the two-planet spring tides of the tidally dominant planets
Venus, Earth and Jupiter are indeed capable of exciting magneto-Rossby waves
with amplitudes of up to m/s, or even more, depending on the widely unknown
damping parameter $\lambda$. This identification of a mechanism by which the
tidal energy can be “harvested” by the Sun reinstates the relevance of the 1
mm tidal height, which has been known for a long time to correspond
energetically to a velocity of 1 m/s (Öpik, 1972), but which was often
(mis-)used to deride any sort of solar-dynamo synchronization.

We went on by asking how the beat periods of those magneto-Rossby waves that
are excited by the two-planets’ spring tides might be capable of synchronizing
a conventional $\alpha-\Omega$ dynamo. While it had been shown previously
(Klevs, Stefani, and Jouve, 2023) that an $\alpha$-effect in the range of dm/s
could lead to parametric resonance of an underlying conventional
$\alpha-\Omega$ dynamo, we focused here on the two most promising mechanisms,
viz, the sensitivity of field storage capacity and helicity oscillations. In
either case we argued that it is the nonlinear (basically squared) action of
the tidally triggered waves that is important, which naturally introduces the
11.07-year beat period period. We have to admit, however, that the exact
mechanism is not yet understood and will require more investigations in the
future. The most obvious next step in this direction is the computation of the
$\alpha$-effect, utilizing the procedures of Avalos-Zuñiga, Plunian and Rädler
(2009).

A similar caveat applies to the second mechanism which, again, tries to
explain long-term periods by the beats of shorter-term periods, this time
focusing on the Suess-de Vries cycle. We have used a slightly enhanced version
of the 2D-dynamo model of Klevs, Stefani, and Jouve (2023) to explain the
latter as a beat period between the 22.14-year Hale cycle and the 19.86-year
period of the rosette-shaped barycentric motion of the Sun.

In our view, this conceptual shift towards first identifying the energy
transfer on the shortest-possible time scales and looking afterwards for beat
periods of the excited waves solves a couple of problems, among them the lack
of direct tidal forcing at the 11.07-yr period (Nataf, 2022; Cionco,
Kudryavtsev, and Soon, 2023) as well as Charbonneau’s noise-related argument
against the viability of a direct planetary forcing on the centennial and
millennial time-scale (Charbonneau, 2022) .

In this respect, our line of argument is quite similar to that of Raphaldini
et al. (2019, 2022) who also tried to explain, first, the Schwabe cycle in
terms of nonlinear triads of magneto-Rossby waves with a ten-times shorter
time-scale, and, second, the Gleissberg and Suess-de Vries cycles as higher
order “precession resonances”, including two such triads. In contrast to these
works, however, we focused here on the very specific 11.07-year period that
shows up in the non-linear interaction of three tidally-induced magneto-Rossby
waves. A second difference applies to the transition from Rieger, via Schwabe,
to Suess-de Vries for which, in our opinion, the argument of Raphaldini et al.
(2022) may run into conceptual problems due to the intervening oscillatory
dynamo field which changes “on-the-fly” the eigenfrequencies of the underlying
magneto-Rossby waves.

By contrast, we relied here on an independent beat mechanism between the
22.14-year Hale cycle and the 19.86-year barycentric motion which has the
absolutely strongest effect on the angular momentum (see Figure 2 in Stefani,
Stepanov, and Weier (2021)), although the specific spin-orbit coupling
mechanism is not well understood yet. Again, our focus was on a sharp 193-year
beat period (as indeed observed by Prasad et al. (2004)) rather than on some
broad signal on the general 100–200-year scale. That said, we believe that the
mathematical framework of triadic (and “precession”) interactions as developed
by Raphaldini et al. (2019, 2022), together with the methods to derive the
$\alpha$-effect for Rossby waves Avalos-Zuñiga, Plunian and Rädler (2009),
represent an ideal starting point for more quantitative corroborations of the
ideas that could only be sketched in this article. Complementarily to this
focus on the $\alpha$-effect, one might also evaluate the non-linear energy
transfer terms from the Appendix of Dikpati et al. (2018) which, via
tachocline nonlinear oscillations, may lead to a similar $\Omega$-related
synchronization mechanism as once proposed by Zaqarashvili (1997).

As for the longest time scales (Eddy, Bray-Hallstatt) we have reiterated the
argument of Stefani, Stepanov, and Weier (2021) that they might well occur as
chaotic breakdowns of the shorter Suess-de Vries cycle, quite in accordance
with the supermodulation concept of Weiss and Tobias (2016). Yet, this does
not exclude some sort of residual regularity, perhaps driven by the 2318-year
cycle of the Jupiter-Saturn-Uranus-Neptune system for which there is also some
observational evidence (McCracken and Beer, 2008).

In view of the growing evidence for a significant role of the Sun for the
terrestrial climate, both on the decadal and centennial (Connolly et al.,
2021; Stefani, 2021; Scafetta, 2023) as well as on the millennial time-scale
(Bond et al., 2001), we consider a deepened understanding of any such kind of
(quasi-)deterministic triggers of the solar dynamo as worthwhile and timely.

## Appendix

In this Appendix, we are concerned with two specific aspects related to the
azimuthal averages carried out in Section 4.

First, we explain why the azimuthally integrated square of the 3-wave-signal
still contains the 11.07-year periodicity (Figure 8a), while that of the
corresponding 2-wave-signal (Figure 8b) does not. For that, we show in Figure
12 the corresponding squares of the signals at different angles $\varphi$
(indicated by the colored numbers). Obviously, for the 3-wave signal (a) the
maximum of the beat for different angles remains at the same instant in time
(reflecting the “orbital invariance” according to Scafetta (2022)), while it
shifts linearly for the 2-wave signal (b). The integration over the angles
explains the differences seen in Figure 8.

Figure 12.: Values of the squares under the integrals of Equation (6) at 7
different angles $\varphi$ (colored number on the r.h.s.), for the entire
3-wave signal (a) and the 2-wave signal without the last term (from Venus-
Earth). The black line links the instants of the maxima at different angles.
While this instant remains the same in (a), it is shifted to later instants in
(b).Fig:fig12

Second, we consider the issue that the dynamo relevant non-linear term does
not correspond exactly to Equation (6) but to a more complicated one including
an additional phase shift $\psi_{0}$ in the second factor, according to

$\displaystyle S_{\rm ps}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\left[\cos\left(2\pi\cdot\frac{t-t_{\rm
VJ}}{0.5\cdot P_{\rm VJ}}+2\varphi\right)+\cos\left(2\pi\cdot\frac{t-t_{\rm
EJ}}{0.5\cdot P_{\rm EJ}}+2\varphi\right)\right.$ (9)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left.\cos\left(2\pi\cdot\frac{t-t_{\rm
VE}}{0.5\cdot P_{\rm VE}}+2\varphi\right)\right]$
$\displaystyle\times\left[\cos\left(2\pi\cdot\frac{t-t_{\rm VJ}}{0.5\cdot
P_{\rm VJ}}+2\varphi+\psi_{0}\right)+\cos\left(2\pi\cdot\frac{t-t_{\rm
EJ}}{0.5\cdot P_{\rm EJ}}+2\varphi+\psi_{0}\right)\right.$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left.\cos\left(2\pi\cdot\frac{t-t_{\rm
VE}}{0.5\cdot P_{\rm VE}}+2\varphi+\psi_{0}\right)\right]$

Such a phase shift would be relevant, in particular, for the azimuthally
averaged helicity wherein - for the case of magneto-Rossby waves - the
vorticity is typically phase shifted by 90∘ with respect to the vertical
velocity.

Although it is rather trivial in view of the integration law of products of
two trigonometric functions, Figure 13 shows how this average converges to
zero with increasing phase shift $\psi_{0}$ (illustrated by $\psi_{0}$ values
of 0∘, 45∘ and 75∘.)

Figure 13.: Signals $S_{\rm ps}$ according to Equation (9) with the phase
shifts $\psi_{0}$ (a) equal to 0∘, (b) equal to 45∘, (c) equal to
75∘.Fig:fig13

### Acknowledgments

This work received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 787544). Inspiring discussions with Carlo Albert, Jürg Beer, Axel
Brandenburg, Robert Cameron, Antonio Ferriz Mas, Günther Rüdiger, Laurène
Jouve, Henri-Claude Nataf, Markus Roth, Dmitry Sokoloff, Rodion Stepanov,
Steve Tobias, and Teimuraz Zaqarashvili on various aspects of the solar
dynamo, and its synchronization, are gratefully acknowledged.

## Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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