Supercriticality of the Dynamo Limits the Memory of the Polar Field to One Cycle

In the flux-transport dynamo model (Charbonneau 2010; Karak et al. 2014), we solve the following equations for axisymmetric magnetic fields:

$$\begin{eqnarray}&&\displaystyle \frac{\partial A}{\partial t}+\displaystyle \frac{1}{s}({\boldsymbol{v}}\cdot {\boldsymbol{\nabla }})({sA})={\eta }_{p}\left({{\rm{\nabla }}}^{2}-\displaystyle \frac{1}{{s}^{2}}\right)A+{S}_{\alpha },\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial B}{\partial t}+\displaystyle \frac{1}{r}\left[\displaystyle \frac{\partial }{\partial r}({{rv}}_{r}B)+\displaystyle \frac{\partial }{\partial \theta }({v}_{\theta }B)\right]={\eta }_{t}\left({{\rm{\nabla }}}^{2}-\displaystyle \frac{1}{{s}^{2}}\right)B\\ \quad +\,s({{\boldsymbol{B}}}_{p}.{\rm{\nabla }}){\rm{\Omega }}+\displaystyle \frac{1}{r}\displaystyle \frac{d{\eta }_{t}}{{dr}}\displaystyle \frac{\partial B}{\partial r},\end{array}\end{eqnarray} \tag{ 3 }$$

where A and B are the potential of the poloidal magnetic field ( B _p ) and the toroidal magnetic field, respectively, such that ${{\boldsymbol{B}}}_{{\rm{t}}{\rm{o}}{\rm{t}}{\rm{a}}{\rm{l}}}={{\boldsymbol{B}}}_{p}+B{\hat{{\boldsymbol{e}}}}_{\phi }$, with ${{\boldsymbol{B}}}_{p}={\boldsymbol{\nabla }}\times A{\hat{{\boldsymbol{e}}}}_{{\boldsymbol{\phi }}}$, $s=r\sin \theta$ with θ being the colatitude, ${\boldsymbol{v}}={v}_{r}{\hat{{\boldsymbol{e}}}}_{{\boldsymbol{r}}}+{v}_{\theta }{\hat{{\boldsymbol{e}}}}_{\theta }$ is the meridional circulation, Ω is the angular velocity, η_p and η_t are the diffusivities of the poloidal and toroidal fields, respectively, and S_α is the parameter that captures the Babcock–Leighton process for the generation of the poloidal field from toroidal one. We use six models, namely Models I, II, III, IV, V, and VI.

2.1.1. Models I–II

For Models I and II, we use the local prescription of α, which was done in the Surya code (Nandy & Choudhuri 2002; Chatterjee et al. 2004; Choudhuri 2018). In these models, S_α = α B, where

$$\begin{eqnarray}\begin{array}{rcl}\alpha & = & \displaystyle \frac{{\alpha }_{0}}{4}\cos \theta \left[1+\mathrm{erf}\left(\displaystyle \frac{r-0.95{R}_{\odot }}{0.025{R}_{\odot }}\right)\right]\\ & & \times \left[1-\mathrm{erf}\left(\displaystyle \frac{r-{R}_{\odot }}{0.025{R}_{\odot }}\right)\right].\end{array}\end{eqnarray} \tag{ 4 }$$

For the present study, we use the same model as given in Yeates et al. (2008). For Model I, we have used the parameters for the poloidal field diffusion: η₂ = 1 × 10¹² cm² s⁻¹ and η₀ = 2 × 10¹² cm² s⁻¹ and for the meridional circulation: v₀ = 15 m s⁻¹, while for Model II, we use the same diffusivities, but v₀ = 26 m s⁻¹. Yeates et al. (2008) call these two models the diffusion-dominated regime (their Run 1) and the advection-dominated regime (Run 2; see their Section 5.1).

2.1.2. Models III–VI

For Models III–VI, we use the nonlocal α prescription, as initially done in Dikpati & Charbonneau (1999), and followed by many authors; see Choudhuri et al. (2005) and Choudhuri & Hazra (2016) for a discussion on the different α prescriptions. Thus, in this model,

$$\begin{eqnarray}&&{S}_{\alpha }=\displaystyle \frac{\alpha }{1+{\left(\tfrac{B(0.7{R}_{\odot },\theta )}{{B}_{0}}\right)}^{2}}B(0.7{R}_{\odot },\theta ),\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}\alpha & = & \displaystyle \frac{{\alpha }_{0}}{4}\sin \theta \cos \theta \left[\displaystyle \frac{1}{1+{e}^{-\gamma (\theta -\pi /4)}}\right]\\ & & \times \,\left[1+\mathrm{erf}\left(\displaystyle \frac{r-0.95{R}_{\odot }}{0.05{R}_{\odot }}\right)\right]\left[1-\mathrm{erf}\left(\displaystyle \frac{r-{R}_{\odot }}{0.01{R}_{\odot }}\right)\right],\end{array}\end{eqnarray} \tag{ 6 }$$

with γ = 30 and B₀ = 4 × 10⁴ G. Also, η_t = η_p = η, where

$$\begin{eqnarray*}\begin{array}{rcl}\eta (r) & = & {\eta }_{\mathrm{RZ}}+\displaystyle \frac{{\eta }_{0}}{2}\left[1+\mathrm{erf}\left(\displaystyle \frac{r-0.7{R}_{\odot }}{0.02{R}_{\odot }}\right)\right]\\ & & +\displaystyle \frac{{\eta }_{\mathrm{surf}}}{2}\left[1+\mathrm{erf}\left(\displaystyle \frac{r-0.9{R}_{\odot }}{0.02{R}_{\odot }}\right)\right],\end{array}\end{eqnarray*}$$

with η_RZ = 5 × 10⁸ cm² s⁻¹ and η_surf = 2 × 10¹² cm² s⁻¹. Basically, we use the same model as given in the Reference Solution of Hotta & Yokoyama (2010) with only a change in the value of η_surf. For Model III, we use η₀ = 5 × 10¹⁰ cm² s⁻¹, while for Model IV, we use five times less diffusivity than in Model III. Model V is the same as Model III but the meridional circulation is switched off. In Model VI, η₀ is varied, but α₀ is fixed at 0.4 ms⁻¹.

Finally, we use a time-delay dynamo model that was developed in Wilmot-Smith et al. (2006); also see Hazra et al. (2014) for an application of this model. In this model, the equations for the toroidal and poloidal fields are truncated by removing the spatial dependences, and some time delays are introduced to mimic the finite times required to communicate the fields between the base of the convection zone (BCZ) to the surface through meridional flow and magnetic buoyancy. The following equations are solved in this model:

$$\begin{eqnarray}&&\displaystyle \frac{{dB}}{{dt}}=\displaystyle \frac{\omega }{L}A(t-{T}_{0})-\displaystyle \frac{B}{{\tau }_{d}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dA}}{{dt}}=\displaystyle \frac{{\alpha }_{0}}{1+{\left[\tfrac{B(t-{T}_{1})}{{B}_{\mathrm{eq}}}\right]}^{2}}B(t-{T}_{1})-\displaystyle \frac{B}{{\tau }_{d}}.\end{eqnarray} \tag{ 8 }$$

Here, T₀ represents the time delay required for the generation of a toroidal field from the poloidal one through differential rotation, while T₁ is the time delay involved in the production of a poloidal field from the toroidal field through the Babcock–Leighton process. ω and L are the contrast in differential rotation and length scale in the tachocline, respectively, τ_d is the diffusion timescale of the turbulent diffusion in the CZ, and α₀ is the amplitude of the α (like the one in flux-transport dynamo models). We note that unlike in the previous delay dynamo model of Wilmot-Smith et al. (2006), here we have included the usual alpha quenching: $1/(1+{(B/{B}_{\mathrm{eq}})}^{2})$. The parameters for our study are as follows: T₀ = 2, T₁ = 0.5, w/L = −0.34, B_eq = 1, α₀ = 0.29, and τ_d = 15.

Let us identify the dynamo transitions in each model. As seen in Equation (1), the growth of the magnetic field is possible when D exceeds a critical value. In our models, this is possible either by increasing α₀ or by decreasing η₀. In Models I–V, we vary α₀ while keeping other parameters the same. Figure 1 shows the mean ϕ_tor from these runs; ϕ_tor is computed over a layer r = 0.677R_⊙–0.726R_⊙ and latitudinal extent 10°–45°, and normalized by the area and B₀. In Model VI, we fix α₀ at 0.4 m s⁻¹ and decrease η₀; Figure 1(c) shows this result. We find that the critical α₀, defined as ${\alpha }_{0}^{\mathrm{crit}}$, for the dynamo transition in Models I, II, III, IV, and V are 6.2, 10, 0.38, 0.05, and 0.8 m s⁻¹. In Model VI, where the critical η₀, defined as ${\eta }_{0}^{\mathrm{crit}}$, is around 0.05 × 10¹² cm² s⁻¹. As long as α₀ is above these critical values (or η₀ below ${\eta }_{0}^{\mathrm{crit}}$), the magnetic field shows a regular polarity reversal. Increasing α₀ (or decreasing η₀) makes the model more supercritical. In the supercritical regime, the magnetic field does not grow linearly because of the nonlinearity imposed in these kinematic dynamo models.

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Let us make a few comments on Figure 1. We observe that in Models III–V, the variation of magnetic field with α₀ for ${\alpha }_{0}\gt {\alpha }_{0}^{\mathrm{crit}}$ is different from that in Models I and II. This difference is because of the different ways of including nonlinearity in these models. In Models I and II (Surya/local α), the nonlinearity is included in the magnetic buoyancy part in the following way. When the toroidal field in any latitude grid above the BCZ at intervals of time 8.8 × 10⁵ s exceeds a certain value (B₀ = 0.8 × 10⁵ G), 50% of the flux at that grid is reduced and the same is deposited at a surface layer. This reduction of toroidal flux is the only nonlinearity in Models I and II. However, in Models III–VI, a nonlinearity of the form $1/[1+{(B/{B}_{0})}^{2}]$ is included in the Babcock–Leighton α term; see Equation (5).

Now we include stochastic fluctuations in the Babcock–Leighton α to produce variable magnetic cycles at different regimes of the dynamo, i.e., at different dynamo parameters. To do so, in Equations (4) and (6), we replace α₀ by α₀(1 + σ(t, τ_cor)), where σ is the uniform random deviation within [−1, 1], and τ_cor is the time after which α₀ is updated. The value of τ_cor is chosen in such a way that the ratio of cycle period to τ_cor remains the same. For reference, τ_cor = 2.3 yr and 1.5 yr, respectively, for Models I and II when α₀ = 30 m s⁻¹. Unless stated otherwise, in all stochastically forced simulations, we include 100% fluctuations. ¹ Thus, the relative fluctuation level is kept the same. We perform simulations at different values of ${\hat{\alpha }}_{0}={\alpha }_{0}/{\alpha }_{0}^{\mathrm{crit}})$ (for Models I–V) and ${\hat{\eta }}_{0}={\eta }_{0}^{\mathrm{crit}}/{\eta }_{0}$ (for Model VI).

We realized when 100% fluctuations in α₀ is added, Models I and II (Surya) tend to decay unless α₀ is sufficiently above ${\alpha }_{0}^{\mathrm{crit}}$. Or in other words, ${\alpha }_{0}^{\mathrm{crit}}$ is increased in Models I and II when fluctuations are included. This does not happen in Models III–V (nonlocal α). This different behavior is due to the extensively different parameters and the treatment of magnetic buoyancy in these models. Therefore, with 100% fluctuations in α₀, we obtain stable cycles in Models I and II only when ${\hat{\alpha }}_{0}\geqslant 2.0$. In Table 1, we present the correlations between the peaks of the polar flux and the peaks of the low-latitude toroidal flux at the base of CZ (ϕ_r is computed on the solar surface over the latitudinal extent 70°–89° and normalized by the area and B₀).

Table 1. Correlation Coefficients between ϕ_r (n) and ϕ_tor of Different Cycles for Simulations at Different Values of ${\hat{\alpha }}_{0}$ and ${\hat{\eta }}_{0}$

Local α (Surya)					Nonlocal α						Nonlocal α
		Model I	Model II				Model III	Model IV	Model V			Model VI
${\hat{\alpha }}_{0}$	ϕr (n) &	r (s.l.?)	r (s.l.?)		${\hat{\alpha }}_{0}$	ϕr (n) &	r (s.l.?)	r (s.l.?)	r (s.l.?)		${\hat{\eta }}_{0}$	ϕr (n) &	r (s.l.?)
2.0	ϕtor (n)	0.94 (Y)	0.96 (Y)		1.0	ϕtor (n)	0.92 (Y)	0.74 (Y)	0.86 (Y)		1.0	ϕtor (n)	0.79 (Y)
	ϕtor (n+1)	0.96 (Y)	0.98 (Y)			ϕtor (n+1)	0.99 (Y)	0.99 (Y)	0.94 (Y)			ϕtor (n+1)	0.99 (Y)
	ϕtor (n+2)	0.90 (Y)	0.96 (Y)			ϕtor (n+2)	0.91 (Y)	0.78 (Y)	0.77 (Y)			ϕtor (n+2)	0.77 (Y)
	ϕtor (n+3)	0.86 (Y)	0.92 (Y)			ϕtor (n+3)	0.89 (Y)	0.68 (Y)	0.60 (Y)			ϕtor (n+3)	0.73 (Y)
2.5	ϕtor (n)	0.71 (Y)	0.69 (Y)		1.5	ϕtor (n)	0.23 (Y)	0.39 (Y)	0.32 (Y)		1.5	ϕtor (n)	0.50 (Y)
	ϕtor (n+1)	0.80 (Y)	0.87 (Y)			ϕtor (n+1)	0.99 (Y)	0.97 (Y)	0.89 (Y)			ϕtor (n+1)	0.99 (Y)
	ϕtor (n+2)	0.54 (Y)	0.79 (Y)			ϕtor (n+2)	0.23 (Y)	0.47 (Y)	0.35 (Y)			ϕtor (n+2)	0.48 (Y)
	ϕtor (n+3)	0.47 (Y)	0.59 (Y)			ϕtor (n+3)	0.26 (Y)	0.42 (Y)	0.20 (Y)			ϕtor (n+3)	0.49 (Y)
3.0	ϕtor (n)	0.40 (Y)	0.59 (Y)		2.0	ϕtor (n)	−0.07 (N)	0.19 (Y)	0.18 (Y)		2.0	ϕtor (n)	0.28 (Y)
	ϕtor (n+1)	0.72 (Y)	0.82 (Y)			ϕtor (n+1)	0.99 (Y)	0.97 (Y)	0.89 (Y)			ϕtor (n+1)	0.99 (Y)
	ϕtor (n+2)	0.26 (Y)	0.40 (Y)			ϕtor (n+2)	−0.09 (N)	0.29 (Y)	0.25 (Y)			ϕtor (n+2)	0.28 (Y)
	ϕtor (n+3)	0.26 (Y)	0.23 (Y)			ϕtor (n+3)	0.18 (N)	0.20 (Y)	0.07 (N)			ϕtor (n+3)	0.41 (Y)
4.0	ϕtor (n)	0.27 (Y)	0.37 (Y)		4.0	ϕtor (n)	−0.39 (Y)	−0.12 (N)	0.02 (N)		4.0	ϕtor (n)	0.03 (N)
	ϕtor (n+1)	0.67 (Y)	0.75 (Y)			ϕtor (n+1)	0.99 (Y)	0.95 (Y)	0.88 (Y)			ϕtor (n+1)	0.99 (Y)
	ϕtor (n+2)	−0.01 (N)	0.10 (N)			ϕtor (n+2)	−0.40 (Y)	0.00 (N)	0.09 (N)			ϕtor (n+2)	0.04 (N)
	ϕtor (n+3)	0.10 (N)	−0.02 (N)			ϕtor (n+3)	0.25 (Y)	0.05 (N)	−0.01 (N)			ϕtor (n+3)	0.33 (Y)
5.0	ϕtor (n)	0.15 (N)	0.31 (Y)
	ϕtor (n+1)	0.72 (Y)	0.68 (Y)
	ϕtor (n+2)	−0.03 (N)	0.03 (N)
	ϕtor (n+3)	0.10 (N)	0.04 (N)

Note. Here, ${\hat{\alpha }}_{0}={\alpha }_{0}/{\alpha }_{0}^{\mathrm{crit}}$ and ${\hat{\eta }}_{0}={\eta }_{0}^{\mathrm{crit}}/{\eta }_{0}$, where the superscript, "crit," represents the minimum (maximum) values of α₀ (η₀) needed for the dynamo action in each model; see Figure 1. In Models I–V, α₀ is increased, while in Model VI, η₀ is decreased in different runs. A total of 275 data (cycles) are used to obtain the correlations. The abbreviation Y or N represents whether the correlation is statistically significant or not based on the computation of the significance level (s.l. = (1 − p)100%). The threshold for significance is 95%. In all simulations, 100% fluctuations in α₀ are included. However, due to statistical uncertainties in numerical realizations, the correlation coefficients are slightly different when runs are repeated with different realizations of random numbers.

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We first consider the results from Models I and II, which are shown on the left columns of Table 1 and Figure 2. We observe that when ${\hat{\alpha }}_{0}$ is small, ϕ_r(n) correlates strongly with ϕ_tor of cycle n, n + 1, n + 2, and n + 3. These multiple-cycle correlations decrease with the increase of ${\hat{\alpha }}_{0}$. In the highly supercritical regime (large ${\hat{\alpha }}_{0}$), ϕ_r(n) strongly correlates with ϕ_tor of cycle n + 1 alone.

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Let us now discuss the physics behind these correlations. Our discussion will be largely based on Jiang et al. (2007), Yeates et al. (2008), and Charbonneau & Barlet (2011). In the Babcock–Leighton models, the toroidal field gives rise to the poloidal field through the α effect (not the mean-field α but a simplified parameterization of the Babcock–Leighton process). In this process, there is nonlinearity and some randomness. The poloidal field generated in the low latitudes is first transported to high latitudes and then to the deep CZ through meridional circulation and diffusion. Finally, this poloidal field gives rise to the toroidal field for the following cycle through differential rotation. Thus, the dynamo chain can be written as follows

$$\begin{eqnarray}&&{\phi }_{\mathrm{tor}}(n)\underset{+\ \mathrm{Nonlinearity}}{\overset{\mathrm{Randomness}}{\longrightarrow }}{\phi }_{{\rm{r}}}(n)\underset{\mathrm{Deterministic}}{\overset{({\rm{\Omega }}\,\mathrm{effect})}{\longrightarrow }}{\phi }_{\mathrm{tor}}(n+1)...\end{eqnarray} \tag{ 9 }$$

Clearly, the correlation between ϕ_r(n) and ϕ_tor(n) is affected by randomness and nonlinearity. When the model operates near the critical α₀, the nonlinearity in α is weak, and thus, the correlation between ϕ_r(n) and ϕ_tor(n) is determined only by the randomness included in the Babcock–Leighton α. The randomness that we have included in our model tries to reduce this correlation and thus the correlation between ϕ_r(n) and ϕ_tor(n) is not perfect even at the smallest α₀ (${\hat{\alpha }}_{0}=2.0;$ Table 1). We emphasize that as long as α₀ is not completely random, there is a significant correlation between ϕ_r(n) and ϕ_tor(n). This is what is seen in Table 1 for ${\hat{\alpha }}_{0}=2$ in Models I and II.

As the dynamo becomes supercritical, the dynamo growth rate and the nonlinearity increase. This nonlinearity, along with the randomness, spoils the linear relation in the chain: ϕ_tor(n) → ϕ_r(n). Hence, the correlation between ϕ_r(n) and ϕ_tor(n) reduces with the increase of α₀. As seen in Table 1, simulations at ${\hat{\alpha }}_{0}\gt 3$ do not show this correlation strongly.

Next, the strong correlation between ϕ_r(n) and ϕ_tor(n + 1) in all the runs is easy to understand. As ϕ_r(n) → ϕ_tor(n + 1) involves only the Ω effect, which is deterministic in our model, the correlation between ϕ_r(n) and ϕ_tor(n + 1) must be strong in all runs. We note that this correlation holds in any model, as long as the poloidal field feeds back to the toroidal one through diffusion and/or advection (Charbonneau & Barlet 2011). This is what is seen in Table 1. There is a slight decrease of this n–(n + 1) correlation with the increase of α₀. This is possibly due to the systematic decrease of the mean cycle period with the increase of α₀; see Table 2. When the cycle period decreases, the transport time of the poloidal field effectively decreases. Thus, the less efficient transport of the poloidal field causes a decrease in the correlation between the polar flux and the next cycle toroidal flux.

Table 2. The Mean Cycle Period T_av (in Years) of the Simulations Presented in Table 1

Model:	I	II			III	IV	V		VI
	(Local α)				(Nonlocal α)				(Nonlocal α)
${\hat{\alpha }}_{0}$	Tav	Tav		${\hat{\alpha }}_{0}$	Tav	Tav	Tav		${\hat{\eta }}_{0}$	Tav
2.0	22.4	13.8		1.0	9.1	9.3	8.3		1.0	9.0
2.5	21.6	13.5		1.5	8.9	9.1	8.4		1.5	9.1
3.0	21.5	13.1		2.0	8.8	8.9	8.5		2.0	9.1
4.0	20.8	12.8		4.0	8.7	8.6	9.0		4.0	9.3
5.0	20.2	12.5

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Finally, the n–(n + 2) and n–(n + 3) correlations are linked to the part ϕ_tor → ϕ_r. If the nonlinearity is weak, then the correlations of ϕ_r(n) with ϕ_tor(n + 2) and ϕ_tor(n + 3) will be determined by the randomness in α and the efficiency of the magnetic field transport, and hence, there will be some correlation. As the dynamo is marginally supercritical at ${\hat{\alpha }}_{0}=2$, multiple-cycle correlations are seen in the top four panels of Figure 2. On the other hand, in the supercritical regime, these multiple-cycle correlations will disappear because the chain ϕ_tor → ϕ_r is spoiled by the nonlinearity and randomness. Thus, all n–(n + 2) and n–(n + 3) correlations are negligible at large ${\hat{\alpha }}_{0};$ see Table 1 and the bottom four panels of Figure 2.

In Figure 3, we observe that when the dynamo is near the critical value, it produces extended episodes of weaker activity, resembling the solar grand minima, and as the supercriticality increases, the frequency of grand minima decreases. In the highly supercritical regime, the model does not produce any grand minima even when the fluctuation level is increased (because the dynamo growth is large and the magnetic field quickly grows from the weaker value). During these grand-minimum phases, the model becomes very linear and causes multicycle correlations in the polar field. In the supercritical regime, as the model does not enter into any extended grand-minimum phase, the model remains nonlinear all the time and thus it cannot produce multicycle correlation. In fact, we have seen that if we exclude these grand-minimum phases from the data of the weakly supercritical regime (red-colored zones in Figure 3), then we observe that the multicycle correlation is reduced to one cycle. This supports our conclusion that when the dynamo is marginally supercritical (weakly nonlinear) and when the α₀ is not completely random, there will be a good correlation between ϕ_tor(n) and ϕ_r(n). When this n–n correlation holds, the dynamo chain will allow ϕ_r(n)–ϕ_tor(n + 2) and ϕ_r(n)–ϕ_tor(n + 3) correlations to persist.

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We recall that both simulations labeled diffusion-dominated and advection-dominated regimes in Yeates et al. (2008) were performed at α₀ = 30 m s⁻¹ (their Runs 1 and 2). They claimed that the one-cycle correlation found in diffusion-dominated simulation is due to the dominant contribution of the diffusion over the advection. We do not agree with this conclusion because if this is the case, then at smaller α₀, the same correlations would also have been observed, which is not the case; see Table 1 and Figure 2. In reality, the supercriticality of their two models is different. As we have shown (Figure 1) that the critical α₀ in their two models are 6.2 and 10 m s⁻¹. Hence, α₀ = 30 m s⁻¹ correspond to ${\hat{\alpha }}_{0}=4.8387$ in Model I (their diffusion-dominated regime) and ${\hat{\alpha }}_{0}=3$ in Model II (their advection-dominated regime). Thus, what Yeates et al. (2008) call diffusion dominated is actually more highly supercritical than their advection-dominated model. We have seen in our study that with the increase of supercriticality, the model becomes more nonlinear (grand minima become very rare; see the Appendix and Figure 7), and the multiple-cycle correlations are reduced to one cycle.

Incidentally, Hazra et al. (2020) did not find multiple-cycle correlations even in their advection-dominated model (see insignificant correlation values for n–n, n–(n + 2), and n–(n + 3) as given in their Tables 1 and 2, last columns)—a clear contradiction to Yeates et al. (2008). The reason for not finding multiple-cycle correlations in Hazra et al. (2020) is that they included a mean field α, in addition to the Babcock–Leighton source for the poloidal field and this additional α possibly made the model considerably supercritical. And, as we have shown that in the supercritical regime, even if the dynamo is advection dominated, the polar field memory is limited to only one cycle.

Karak & Nandy (2012) showed that when downward turbulent pumping in the poloidal field is included in the advection-dominated dynamo, the multiple-cycle correlation is reduced to one cycle. Actually, when pumping is included in the model, the dynamo becomes strong (because of the suppression of magnetic flux through the surface; see Cameron et al. 2012; Karak & Cameron 2016; Karak & Miesch 2018), and the dynamo transition happens at smaller α₀. Thus, the same model that produced multiple-cycle correlations because of weak supercriticality becomes heavily supercritical with the inclusion of turbulent pumping. This is the hidden reason for a one-cycle correlation being displayed in the advection-dominated model with turbulent pumping.

We do not mean that the turbulent transports, namely, the turbulent diffusivity, pumping, and meridional flow, do not play any role in determining these correlations. The transport is necessary to connect the spatially segregated source regions of the poloidal and the toroidal fields. Through various transport processes, the polar field of a cycle (ϕ_r(n)) is strongly correlated with the amplitude of the next cycle's toroidal field (ϕ_tor(n + 1)). The generation of poloidal field and thus the connection between ϕ_tor(n) and ϕ_r(n) is also possible through turbulent transport. However, as we do not change any parameter of diffusion and meridional flow in each set of runs, our correlations are not affected by turbulent transport processes.

Now we examine the results from Models III and IV (Table 1). These models produce cycles even at ${\hat{\alpha }}_{0}=1$. This is possibly due to much smaller diffusivity and different α prescriptions. We observe that the n–(n + 1) cycle correlation is very strong in all the runs. For Models I and II, this was not the case, because the diffusivity of the poloidal field in the CZ was about 20 times higher than that in Models III and IV. This higher diffusion in Models I and II allowed some of the fluctuations in α to propagate to the same cycle toroidal flux. Despite these differences, we find similar behavior with the increase of ${\hat{\alpha }}_{0}$. That is, when the dynamo is only marginally supercritical (${\hat{\alpha }}_{0}$ near 1), multiple-cycle correlations exist. On the other hand, when the dynamo is supercritical, only a one-cycle correlation holds. This conclusion remains unaffected even when we increase the level of fluctuations. Table 3 shows the results at 150% and 200% fluctuation levels. If we increase the fluctuation levels too much, then the model tends to decay, particularly Models I and II in the weakly supercritical regime.

Table 3. Same as Table 1 but Only for Model IV and at Increased Fluctuation Levels

Nonlocal α
		150%	200%
${\hat{\alpha }}_{0}$	ϕr (n) &	r (s.l.?)	r (s.l.?)
1.0	ϕtor (n)	0.80 (Y)	0.83 (Y)
	ϕtor (n+1)	0.97 (Y)	0.98 (Y)
	ϕtor (n+2)	0.79 (Y)	0.85 (Y)
	ϕtor (n+3)	0.71 (Y)	0.82 (Y)
1.5	ϕtor (n)	0.40 (Y)	0.35 (Y)
	ϕtor (n+1)	0.97 (Y)	0.96 (Y)
	ϕtor (n+2)	0.48 (Y)	0.42 (Y)
	ϕtor (n+3)	0.43 (Y)	0.37 (Y)
2.0	ϕtor (n)	0.30 (Y)	0.33 (Y)
	ϕtor (n+1)	0.96 (Y)	0.96 (Y)
	ϕtor (n+2)	0.36 (Y)	0.42 (Y)
	ϕtor (n+3)	0.31 (Y)	0.35 (Y)
4.0	ϕtor (n)	0.03 (N)	−0.08 (N)
	ϕtor (n+1)	0.92 (Y)	0.91 (Y)
	ϕtor (n+2)	0.14 (N)	0.07 (N)
	ϕtor (n+3)	0.12 (N)	0.18 (N)

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We note that even the model without meridional flow, Model V, also shows the same behavior of the polar field with the increase of the supercriticality; see Table 1. This confirms that advection plays no role in setting the memory of the polar field beyond one cycle, which is in contradiction to Yeates et al. (2008).

Finally, we observe the results from Model VI. As noted earlier, the set of runs in Model VI, is the same as that in Model III, except α₀ is fixed at 0.4 m s⁻¹ and η₀ is decreased (or ${\hat{\eta }}_{0}={\eta }_{0}^{\mathrm{crit}}/{\eta }_{0}$ is increased) in each run. When η₀ is decreased, the model becomes less diffusion dominated. Interestingly, even when the model becomes less diffusive, we observe the shorter of memory of the polar flux. In Table 1, we observe that at smaller ${\hat{\eta }}_{0}$ (1 and 1.5), multiple-cycle correlations hold, while at larger ${\hat{\eta }}_{0}$, only n–(n + 1) correlation holds. The correlation plots for marginally supercritical (${\hat{\eta }}_{0}=1$) and highly supercritical (${\hat{\eta }}_{0}=4$) are shown in Figure 4.

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