A Synchronized Two-Dimensional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}α–\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega $\end{document}Ω Model of the Solar Dynamo

We consider a conventional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}α–\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega $\end{document}Ω-dynamo model with meridional circulation that exhibits typical features of the solar dynamo, including a Hale-cycle period of around 20 years and a reasonable shape of the butterfly diagram. With regard to recent ideas of a tidal synchronization of the solar cycle, we complement this model by an additional time-periodic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}α-term that is localized in the tachocline region. It is shown that amplitudes of some decimeters per second are sufficient for this \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}α-term to become capable of entraining the underlying dynamo. We argue that such amplitudes of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}α may indeed be realistic, since velocities in the range of m s−1 are reachable, e.g., for tidally excited magneto–Rossby waves. Supplementary Information The online version contains supplementary material available at 10.1007/s11207-023-02173-y.

Since even such a remarkable agreement between the average values of two periods might still be a pure coincidence, the question of whether there is a phase coherence between the two time series becomes of utmost importance.The possible phase stability of the Schwabe cycle was first discussed in the paper "Is there a chronometer hidden deep in the Sun?" by Dicke (1978).Analyzing the ratio between the mean square of the residuals (i.e., the distances between the instants of the actual cycle maxima and the hypothetical maxima according to a linear trend) to the mean square of the differences between two consecutive residuals, Dicke's conclusions favoured a clocked process over a random walk process.However, apart from the poor statistics connected with the mere 25 maxima taken into account, one should also take seriously Hoyng's later warning (Hoyng, 1996) that any α-quenching mechanism could easily lead to a sort of self-stabilization of the solar dynamo, making a genuine random walk process "disguise" itself as a clocked process -at least for some centuries.A complementary type of cycle stability appears as a typical feature of conventional Babcock-Leighton dynamos whose period is largely determined by the turnover time of the meridional circulation (Dikpati and Charbonneau, 1999;Charbonneau and Dikpati, 2000;Charbonneau, 2020), which is indeed assumed to be much less fluctuating than the α effect in the convection zone.
With those caveats in mind, we had recently re-considered (Stefani et al., 2020b) the longer time series of cycle minima/maxima as bequeathed to us by Schove (1983), and matched them with two series of the cosmogenic isotopes 10 Be and 14 C. Apart from the hardly decidable existence, or not, of two "lost cycles" (or phase jumps) around 1563 (Link, 1978) and 1795 (Usoskin et al., 2002), our analysis confirmed, by and large, Dicke's conclusion in favour of a clocked cycle, now throughout the last millennium.This result was then put into the context of the most remarkable, though widely overlooked, work of Vos et al. (2004) whose analysis of two series of algae-related data from 10000-9000 cal.BP had evidenced a phase-stable Schwabe cycle with a period of 11.04 years.
In view of those two independent thousand-year long segments showing nearly identical Schwabe cycles with average periods between 11.04 and 11.07 years and the strong evidence for phase stability in either case, we consider it at least worthwhile to quest for a possible physical mechanism that could be capable of linking the weak tidal forces as exerted by planets with the solar dynamo.Setting out from the numerical observation (Weber et al., 2013(Weber et al., , 2015;;Stefani et al., 2016) that a tide-like influence (with its typical m = 2 azimuthal dependence) can entrain the helicity oscillation of an underlying m = 1 instability (the Tayler instability (Tayler, 1973;Seilmayer et al., 2012) for that matter) with barely changing its energy content, we have pursued some rudimentary synchronization studies in the framework of simple 0D and 1D α − Ω-dynamo models (Stefani et al., 2017(Stefani et al., , 2018;;Stefani, Giesecke and Weier, 2019).Within the same framework, we recently tried (Stefani et al., 2020a;Stefani, Stepanov and Weier, 2021) to explain also the longer term Suess-de Vries cycle in terms of a beat period (Wilson, 2013;Solheim, 2013) between the fundamental 22.14-year Hale cycle and the 19.86-yr period of the Sun's barycentric motion (forced, in turn, by the SOLA: klevs.tex;16 January 2023; 2:53; p. 2 orbits of Jupiter and Saturn (Cionco and Pavlov, 2018)) .With the intervening spin-orbit coupling remaining poorly understood, we took resort to the same buoyancy instability mechanism as it had been been employed by Abreu et al. (2012) to plausibilize typical modulation periods on the centennial time-scale.Yet, this similarity between the final results notwithstanding, the fundamental time-scales of our model (22.14 and 19.86 years) that generate the much longer beat period of 193 years, are still close to the period of the undisturbed dynamo.Our mechanism for explaining long-term modulations might, therefore, be less vulnerable to stochastic noise than what was discussed by Charbonneau (2022) in relation to the original model of Abreu et al. (2012).
Admittedly, being restricted to the latitudinal coordinate, our simple 1D dynamo model did not have the requisite level of detail to give a quantitative answer to Charbonneau's recent question of "what, then, can be considered a physically reasonable amplitude for external forcing" (Charbonneau, 2022).It was all the more encouraging that, utilizing a 2D Babcock-Leighton model with a periodic perturbation of the lower operating field threshold of the source term, Charbonneau (2022) found a similarly robust synchronization mechanism as Stefani, Giesecke and Weier (2019).Such a variation of the lower operating field threshold would correspond to variations of the field loss term κ as employed in (Stefani et al., 2020a;Stefani, Stepanov and Weier, 2021) to parameterize the spin-orbit coupling with its 19.86-yr periodicity.While we do not exclude a viable physical translation of the (11.07-yr periodic) tidal forcing into such a type of variation of the field storage capacity, in this paper we will stick to our original idea that it is essentially the α effect that is affected by the tides.Specifically, we seek to know then how much of this periodic α variation would be needed to accomplish synchronization of an otherwise conventional α − Ω-dynamo.Guided by a rough estimation based on the virial assumption U pot ≈ E kin , we consider approximately 1 m/s an upper limit for the tide-induced velocity variation.Given that the value of α, which reflects only the helical part of the turbulence, is typically one order of magnitude lower than the underlying velocity, the focus of our modelling will be on whether α-values of the order of dm/s are sufficient to entrain the entire solar dynamo.
To answer this very specific question, we step back from the more sophisticated double-synchronization model of Stefani et al. (2020a); Stefani, Stepanov and Weier (2021) and restrict ourselves to the very basic tidal synchronization of the Schwabe/Hale cycle.In the next section, we present a rather conventional two-dimensional α − Ω-dynamo with meridional circulation u p , utilizing observation-constrained values for Ω and u p , and employing more or less realistic values of α and the magnetic diffusivity η.To keep the model simple, no specific Babcock-Leighton source term is added to the α-effect "living" in the convection zone.In the next section, we first adjust the value of η to provide a reasonable natural period of the undisturbed dynamo.While the most simple form of the α − Ω model leads, as usual, to a badly shaped butterfly diagram, the correct butterfly shape is recovered by switching on the meridional circulation.Based on the reference model thus defined, we will then assess in detail how much α variation in the tachocline region is actually needed for synchronization.
The paper will conclude with a short discussion of the results and some prospects for future work.

The model
In this section, we motivate and describe our mean-field solar dynamo model and discuss its numerical implementation.Considering only axi-symmetric solutions, we work with a system of partial differential equations whose spatial variables are the co-latitude and the radius.Intentionally, the model has been kept similarly simple as the benchmark model of Jouve et al. (2008).
As usual, the magnetic field is split into a poloidal component B P (r, Θ, t) = ∇ × (A(r, Θ, t)e φ ) and a toroidal component B T (r, Θ, t) = B(r, Θ, t)e φ .The main sources of dynamo action are the gradient of the angular velocity Ω and the α-effect resulting from the helical part of the turbulence in the convection zone.While our model is not a Babcock-Leighton model (which would require a particular source term at the surface) it is a flux-transport model in that it comprises a meridional circulation u p , mainly to ensure a realistic shape of the butterfly diagram.
Choosing the solar radius R = 695700 km as the length and the diffusive time R 2 /η t as the time scale, we employ here -as in Jouve et al. (2008) the dimensionless form of the coupled induction equations for the azimuthal components B := B φ of the magnetic field and A := A φ of the vector potential, wherein we use the notations D 2 := (∇ 2 − s −2 ), s := r sin θ and η = η/η t , with η t being the turbulent magnetic diffusivity in the convection zone.This system is governed by four magnetic Reynolds numbers characterizing, respectively, the effects of shear, meridional circulation, and two different α terms: Herein, Ω eq = 2π × 456 nHz is the angular velocity at the equator, and u 0 and α c max and α p max are the typical intensities of the meridional circulation and the two separate α effects in the convection zone and in the tachocline region.In contrast to Guerrero and de Gouveia Dal Pino ( 2007 We suppose the turbulent magnetic diffusivity η t in the convection zone to be dominated by a strong β effect, whereas it is much smaller in the relatively quiet tachocline region.Refraining from more complicated structures of η as employed, e.g., in Guerrero and de Gouveia Dal Pino (2007) or Sanchez et al. (2014), we use here the simple form of Jouve et al. ( 2008) with η c = 0.01η t , r c = 0.7 and d = 0.02, which shows a smoothed-out jump (by a factor of 100) between the radiation zone and the convection zone.
For the angular velocity we apply the same spatial structure as in Jouve et al. (2008): with r c = 0.7, d = 0.02, Ω c = 0.92 and c 2 = 0.2 (see Figure 1(a)).
For the meridional circulation we chose, again as in Jouve et al. (2008), one single cell defined by u p = ∇ × (ψ(r, Θ)e φ ) with the stream function with r b = 0.65 (see Figure 1(b)).We are well aware of the fact that the specific structure of u p is much less settled than that of Ω(r, Θ), and that more complicated two-cell flows (Kosovichev et al., 2022) might also be considered in future improvements of our model.Finally, α = α c + α p is thought to consist of a conventional part α c in the convection zone, whose time-dependence stems only from the quenching by the magnetic field, SOLA: klevs.tex;16 January 2023; 2:53; p. 5 with B 0 = 1, and an explicitly time-dependent (with forcing period T f ) part α p that is concentrated in the tachocline region, where r d = 0.75.Note that the factor on the second line of Eq. ( 11) represents a resonance term as introduced in Stefani et al. (2016) in order to account for a field-dependent optimal reaction of the underlying instability (e.g., Tayler instability) on the tidal forcing.A similar field dependence has been used, e.g., in Charbonneau (2022), although with the slightly different interpretation as a nonlinearity of the non-local source term that incorporates both a lower and upper operating threshold on the strength of the toroidal magnetic at the base of the convection zone.The spatial structures of these two α terms are visualized in Figure 1(c,d), in either case disregarding any magnetic-field dependence.
For the numerical solution, an explicit finite difference scheme in two dimensions in spherical coordinates is used.As in Rüdiger et al. (2003), the standard resolution was 64 × 64 grid points in both radial and latitudinal directions.The equations are solved with perfect conductor boundary conditions A = ∂(rB)/∂r = 0 at r = 0.65R and vertical field conditions B φ = B Θ = 0 at r = R .

Results
In this section we present and assess the results of three dynamo models with increasing complexity.

Non-synchronized model, without meridional circulation
First we consider the simplest case of a Parker's migratory dynamo (Parker, 1955), without any synchronization term (α p = 0), and without meridional circulation (u p = 0).For the sake of concreteness, we set η t = 2.13 × 10 11 cm 2 /s, and α c max = 1.30m/s, which both are close to the respective geometric means of the lower and upper values as typically found in the literature (10 10 − 10 13 cm 2 /s for η and 10 − 10 3 cm/s for α, see Charbonneau ( 2020)).The resulting magnetic Reynolds numbers according to Equations ( 3) and ( 5) are C Ω = 65100 and C c α = 42.46.The radial dependencies of η(r) and α c (in its unquenched form) are illustrated, for Θ = 45 • , in Figure 2(a).Note that at this particular angle α c (r) does not reach the maximum value of 1.30 m/s.
Figure 3 illustrates the resulting field dependence on time and latitude, taken at r = 0.95, showing a reasonable dynamo cycle period of T d = 14.27 years (i.e.0.0198 diffusion times), but a badly shaped butterfly diagram.

Non-synchronized model, with meridional circulation
In order to recover the correct shape of the butterfly diagram, we switch on a meridional circulation, setting its value to u 0 = 5.2 m/s, which corresponds to R m = 170.For this value, as well as for R m = 200 and 240, the radial dependence of u Θ is shown, again for Θ = 45 • , in Figure 2(b).While the values u Θ at r = 1 are by factor of appr.two too low compared with observations, the typical values of 1-2 m/s at the base of the convection zone are quite compatible with values from helioseismology.Actually, the latter velocities are the crucial ones to set the cycle period.
As seen in Figure 4 we obtain now a butterfly diagram of rather decent shape, and a slightly changed cycle period of T d = 22.798 years.This will serve in the fol-SOLA: klevs.tex;16 January 2023; 2:53; p. 7 lowing as the reference dynamo model whose synchronization is to be evaluated thereupon.While further improvements of the spatio-temporal features of the magnetic field are certainly possible (for example, when including an appropriate Babcock-Leighton source term), we refrain from any further sophistication of the model.

Synchronized model
Finally, we switch on the periodic α term with an assumed forcing period of T f = 11.00 years (we do not insist here on the precise value of 11.07 years).The radial dependence of α p is illustrated by the red curve in Figure 2(a).Note, however, that here α p max has the same value of 1.30 m/s as the corresponding α c max , and that the field-dependent resonance term in Equation ( 11) is artificially set to 1.In reality, the resonance term reduces this value by a factor of 2 for the optimum field strength, and even more so outside the optimum.
As shown in Figure 5, for the specific value α p max = 0.52 m/s we obtain now the dynamo period T d = 22.00 years which corresponds to twice the period T f of the forcing.
In Figure 6 we plot the dependence of the dynamo period T d on α p max .Here we have used a couple of ratios of the "natural" period T n (of the non-synchronized dynamo with α p max = 0) to the forcing periods T f by simply changing the amplitude of meridional circulation which governs T n .Very similar to Figure 10 in Stefani, Giesecke and Weier (2019), and to Figure 10 in Charbonneau (2022), we obtain a clear parametric resonance for some critical value of α p max that depends on the initial distance between twice the forcing period T f and the natural period T n of the unperturbed dynamo.As we had chosen α c max = 1.30m/s, synchronization occurs for an amplitude of α p max in the range of some dm/s.The relative smallness of this number is, of course, a consequence of the 100 times smaller value of η in the tachocline region which amplifies correspondingly the induction effect of α p , even if the latter is concentrated in a significantly smaller zone than α c .That said, we must also admit that synchronization requires a certain proximity of 2T f and T n ; for the R m values indicated by the dashed lines in Figure 6 no clear synchronization was observed even for the highest considered value of α p max /α c max = 1.This narrowness of the synchronizability region, which somewhat contrasts with the broader region obtained in frame of the 1D model (Fig. 10 of Stefani, Giesecke and Weier ( 2019)), might have to do with the tight scaling of T n with the period of the meridional circulation.

Conclusions
As a sequel to the 0D and 1D modelling of solar cycle synchronization (Stefani et al., 2016(Stefani et al., , 2018;;Stefani, Giesecke and Weier, 2019;Stefani et al., 2020a;Stefani, Stepanov and Weier, 2021), we have investigated a more realistic 2D α − Ω-dynamo model.Starting from a conventional set-up without meridional circulation, exhibiting a badly shaped butterfly diagram, via an enhanced model with meridional circulation showing the correct butterfly shape, we have assessed the synchronization capabilities of a time-periodic α term concentrated in the tachocline region.For rather standard values of all other parameters, it was shown that synchronization starts already for a magnitude of this additional α-term as low as some dm/s.The smallness of this value relies on the fact that η in the quiet tachocline region is significantly lower than in the convection zone where it is dominated by the turbulent β effect.The utilized tachoclinic diffusivity η ≈ 2.13 × 10 9 cm 2 /s should be considered a conservative choice; in view of much lower values such as, e.g.2.2 × 10 8 cm 2 /s as used by Guerrero and de Gouveia Dal Pino (2007), the real value of α, required for synchronization, might still be lower than the one derived here.This brings us back to Charbonneau's "elephant in the room: what, then, can be considered a physically reasonable amplitude for external forcing?" (Charbonneau, 2022).Let us recall the very rough energetic consideration Öpik (1972) that the typical tidal height of h tidal = GmR 2 tacho /(g tacho d 3 ) ≈ 1 mm corresponds energetically to a velocity scale of v 0 ∼ (2g tacho h tidal ) 1/2 ≈ 1 m/s when employing the huge gravity at the tachocline of g tacho ≈ 500 m/s 2 .Invoking the equally rough estimate α ∼ v 0 from renormalization theory (Moffatt and Dormy, 2019) (and even when realistically assuming α to be one or two orders of magnitude smaller than v 0 ), a tidally generated α-value of a few dm/s seems not out of reach.Indeed, it was recently shown (Horstmann, 2022) that (magneto-)Rossby waves (Marquez-Artavia, Jones, and Tobias,, 2017;Zaqarashvili, 2018;Dikpati et al., 2020) under the influence of a realistic tidal forcing are capable of acquiring velocity scales of up to 1 m/s.Therefore, it appears that the "astrological homeopathy" (Charbonneau, 2022) of tidal forcing may well be suited to generate an α-effect in the tachocline region that is strong enough to entrain the entire solar dynamo.
We have further confirmed the prior results of Stefani, Giesecke and Weier (2019) (Figure 10) and Charbonneau (2022) (Figure 10) that this type of synchronization requires a certain proximity of the tidal forcing's period to half the "natural" period of the undisturbed dynamo.The Sun, therefore, may just be in the lucky situation of being orbited by a Jupiter with a period that fits nicely to half the "natural" period of the undisturbed dynamo.It remains to be seen whether some peculiar features of the solar dynamo, e.g its somewhat unusual cycle period (Böhm-Vitense, 2007) and, in particular, "its comparatively smooth, regular activity cycle" (Radick et al., 2018), could find an explanation at this point.SOLA: klevs.tex;16 January 2023;2:53;p. 10 What are the next steps to be taken?First and foremost, the specific action of m = 2 tidal forces on various m = 1 instabilities (e.g., Tayler) or waves (e.g., magneto-Rossby), and on the α effect connected with them, has to be quantified in a reliable manner.Complementary work on tidal influences on Rayleigh-Bénard convection, and its large-scale circulation (Stepanov and Stefani, 2019;Jüstel et al., 2020Jüstel et al., , 2022)), might be helpful to elucidate helicity entrainment in a more generic sense.
Second, the possible role of further axisymmetric induction effects, beyond the α effect, has to be clarified.The basic idea of a torque-influenced magnetic buoyancy instability within the tachocline (Ferriz Mas, Schmitt, and Schüssler, 1994;Zhang et al., 2003;Abreu et al., 2012) might play a central role here.It was indeed employed as the basic synchronization mechanism by Charbonneau (2022), while in Stefani et al. (2020a); Stefani, Stepanov and Weier (2021), we had used it only to bring into play the second fundamental period 19.86 years via spin-orbit coupling (yet poorly understood, but see Javaraiah (2003); Shirley (2006); Sharp (2013) for first estimates).It certainly needs much more work to disentangle these two effects.Further to this, we should not overlook alternative axisymmetric (m = 0) instabilities, the possible relevance of which had been discussed by several authors (Dikpati et al., 2009;Rogers, 2011).The recently discovered helical magnetorotational instability for flows with positive radial shear (Mamatsashvili et al., 2019) might be an particularly interesting candidate in this respect.

Figure 2 .
Figure 2. Radial dependence of various dynamo ingredients in physical units, all taken at Θ = 45 • .(a) Diffusivity η(r) (black), α c (r) in the unquenched form (violet), and α p (r) for α p max = α c max and with the field-dependent resonance factor artificially set to 1 (red).(b) u Θ (r) resulting from the stream function of Eq. (9) for three different Rm.

Figure 5 .
Figure 5. Same as Figure 4, but with synchronization by a periodic α-term with amplitude α p max = 0.52 m/s and period T f = 11.00 years.

Figure 6 .
Figure 6.Ratio of the period T d of the signal to the period T f of the forcing in dependence on the relative strength of the forcing α p max /α c max .The color coded curves refer to different ratios of the "natural" period Tn of the non-synchronized dynamo to T f , which has been varied by changing the magnetic Reynolds number Rm of the meridional circulation.Tn can be read off from the value at the ordinate axis multiplied by 11 years; it amounts, for example, to 23.3 years for Rm = 150, to 21.6 years for Rm = 200, and to 19.5 years for Rm = 250.