he origin of grand minima in the sunspot cycl

One of the most striking aspects of the 11-year sunspot cycle is that there have been times in the past when some cycles went missing, a most well-known example of this being the Maunder minimum during 16451715. Analyses of cosmogenic isotopes ($^{14}$C and $^{10}$Be) indicated that there were about 27 grand minima in the last 11,000yr, implying that about 2.7% of the solar cycles had conditions appropriate for forcing the Sun into grand minima. We address the question how grand minima are produced and specifically calculate the frequency of occurrence of grand minima from a theoretical dynamo model. We assume that fluctuations in the poloidal field generation mechanism and in the meridional circulation produce irregularities of sunspot cycles. Taking these fluctuations to be Gaussian and estimating the values of important parameters from the data of last 28 solar cycles, we show from our flux transport dynamo model that about 14% of the sunspot cycles may have conditions suitable for inducing grand minima.

A few years after the initiation of telescopic observations of sunspots in 1610, there was a period from 1645 to 1715 when very few sunspots appeared on the face of the Sun. This period is known as the Maunder minimum @eddy. Although reliable sunspot data did not exist before 1610, the solar activity at earlier times can be studied from the analyses of the abundances of cosmogenic isotopes like $^{14}$C in old tree rings @solanki04 [@nagaya] and $^{10}$Be in polar ice @beer90. When sunspots are absent, the magnetic field in the solar wind becomes weak and more galactic cosmic rays can reach the Earth, producing more of such radioactive isotopes in the atmosphere. Analyses of these isotopes indicate that there have been about 27 grand minima of different durations in the last 11,000 years @usos07. Since there were about 1,000 solar cycles during this period, the occurrence of 27 grand minima implies that about 2.7% cycles had conditions appropriate for forcing the Sun into grand minima. Furthermore this study showed that the Sun was in the grand minima state for about 17% of the time. We also mention that there are evidences that some solar-like stars show grand minima @baliu. Therefore it is very important to understand the physics of the origin of the grand minima and the probability of occurrence of such grand minima. Moreover, sunspot cycles have an important effect on the space environment and the Earth climate system @lean95 [@crowley00]. Therefore, understanding the grand minima is also important to the space weather and Earth climate communities.

It appears that the most promising theoretical model for the solar cycle at the present time is the flux transport dynamo model @csd95 [@dc99; @nc02], which has been reviewed recently by Charbonneau @c10 and Choudhuri @c11. Let us consider the question how irregularities arise in solar cycles. One important assumption of the flux transport dynamo model is that the poloidal field is produced by the Babcock–Leighton mechanism, in which tilted bipolar sunspots give rise to the poloidal field after their decay. The amount of poloidal field generated depends on the tilt angle of the bipolar sunspots. This tilt is produced by the action of the Coriolis force acting on the magnetic flux tube rising through the solar convection zone due to magnetic buoyancy @dc93 and the average tilt at a solar latitude is given by Joy's law. However, there is a large scatter in the tilt angles around the average given by Joy's law @dasi10, presumably due to the effects of turbulence on the rising flux tubes @lc02. Hence the Babcock–Leighton mechanism has an inherent randomness because of which the poloidal field generated at the end of a cycle would vary from one cycle to the other @ccj07. The second source of irregularities in solar cycles comes from the fluctuations in the meridional circulation of the Sun, which plays a crucial role in the flux transport dynamo. The period of the cycle in the theoretical model is approximately inversely proportional to the amplitude of the meridional circulation @dc99 [@ynm08]. Presumably the variations in the periods of past cycles were produced primarily by variations in the meridional circulation. Although we have direct measurements of the meridional circulation flow speed only during the last few years @hr10 [@ulrich10], the periods of past cycles can be used to draw inferences about meridional circulation variations in the past @k10 [@kc11; @pl08; @lp09]. Such variations in meridional circulation also do cause variations in the strengths of different cycles in addition to variations in their periods. Suppose the meridional circulation has become weaker than usual. Then the period of the cycle will be longer and diffusion will have more time to act on the magnetic field, making the cycle weaker. If diffusion is assumed in the range $10^{12}$–$10^{13}$ cm$^2$ s$^{-1}$ consistent with mixing length arguments @pa79 (which we do in our model), then this effect overcomes the opposing effect of differential rotation also getting more time to generate more toroidal field and the cycles become weaker when the meridional circulation slows down @ynm08 [@k10; @kc11]. However, in the model developed by the HAO group, the diffusion is taken to be about 50 times smaller than what we take @dc99 [@dg06], leading to the opposite effect of cycles getting stronger with slower meridional circulation due to the generation of more toroidal field by differential rotation over a longer time. Several arguments in favor of the higher diffusivity used by us are summarized in Sect. 5 of Jiang et al. @jcc07.

In order to model grand minima theoretically, we have to run our theoretical flux transport dynamo model with fluctuations in poloidal field generation and fluctuations in meridional circulation. Some studies have shown that large fluctuations in poloidal field generation mechanism or large fluctuations in the meridional circulation can force the dynamo into intermittencies resembling grand minima @c92 [@char04; @ck09; @usoskin09; @pl11; @pcb12]. Karak @k10 found that the dynamo is driven into a grand minimum if the poloidal field at the end of a cycle and the meridional circulation at that time fall to sufficiently low values. Defining a grand minimum as absence of sunspots for at least 20 years (the same definition as used by Usoskin et al. @usos07 in estimating the number of grand minima in the past 11,000 years), the shaded region of Fig. 1 indicates the combined values of poloidal field and meridional circulation at the end of a cycle necessary for forcing the dynamo into grand minima according to our theoretical model. The important question now is to estimate the probability of this happening at the end of a cycle. We estimate this probability in the following way.

![The solid line shows the values of the  amplitude $v_0$ and the poloidal field scale factor $\gamma$ which produce grand minima of duration $\sim 20$ yrs. The parameters lying in the shaded region produce grand minima of longer duration. The dashed curves are the contours of the joint probability $P(\gamma, v_0)$, with the values of $P(\gamma, v_0)$ (excluding the constant pre-factor) given in the plot. The scale factor $\gamma$ is defined as the amplitude of poloidal field at the end of a cycle divided by its value in the absence of fluctuations. This figure is produced by using our theoretical solar dynamo model (details given in Karak @k10).<span data-label="parameters"></span>](fig1.eps){width="55.00000%"}

![(a) Histogram of the  amplitude derived from the observed periods of last 28 solar cycles. The solid curve is the Gaussian function with mean = 23 m s$^{-1}$ and standard deviation $\sigma_v$ = 3.34 m s$^{-1}$. (b) Histogram of poloidal field scale factor derived from the peak sunspot number of last 28 solar cycles. The solid curve is the Gaussian function with mean = 1 and standard deviation $\sigma_{\gamma}$ = 0.35. The probabilities plotted along the vertical axis in both (a) and (b) are obtained by dividing the number of data points in a data bin by the total number of data points and the horizontal width of the data bin.<span data-label="hist"></span>](fig2.eps){width="55.00000%"}

Assuming that the inverse of the cycle period gives an approximate value of the meridional circulation amplitude during that cycle (as suggested by theoretical flux transport dynamo models @dc99 [@ynm08]), Karak & Choudhuri @kc11 concluded that the meridional circulation has changed randomly in the past with a correlation time around 30–40 yr (also see the similar study based on the low order dynamo model @pl08 [@lp09]). Figure 2a shows a histogram of the estimated values of the  amplitude during the last 28 cycles. The solid curve in Fig. 2a shows the Gaussian having the mean $\overline{v_0} =
23$ m s$^{-1}$ and the standard deviation $\sigma_v = 3.34$ m s$^{-1}$ calculated from the data presented in the histogram. We find that the Gaussian is a reasonable fit, although we do not have any data points lying way out in the Gaussian tail. Jiang et al.  @jcc07 (also see @dasi10) pointed out on the basis of observational data that there is a good correlation between the  at the end of a cycle and the strength of the next cycle. Assuming a perfect correlation, we can use the strengths of following cycles to obtain values $\gamma$ of  at the ends of previous cycles (scaled by taking their average value as the unit). Figure 2b is a histogram of the values $\gamma$ of  obtained in this way. It should, however, be kept in mind that variations in the  also contribute to fluctuations of cycle strengths @baumann04 [@ynm08; @k10; @kc11] and fluctuations of  at the end of cycles obtained by our method (as shown in Fig. 2b) is probably an over-estimate. The solid curve in Fig. 2b is the Gaussian having the standard deviation $\sigma_{\gamma} 
= 0.35$ obtained from the data in the histogram (the mean is 1 by definition). We expect on general grounds the distribution of $\gamma$ to follow a Gaussian, although we find that the Gaussian fit in Fig. 2b is not as good a fit as in Fig. 2a. This is not surprising given that our data set comprising of only 28 cycles is quite small. If we assume that the fluctuations in the amplitude $v_0$ of  at the solar surface and the fluctuations in $\gamma$ both follow Gaussian distributions, then we can draw one obvious inference. The joint probability that the  at the end of a cycle lies in the range $\gamma$, $\gamma + d\gamma$ and the amplitude of  at the same time lies in the range $v_0$, $v_0 +dv_0$ is given by $$P (\gamma, v_0) d\gamma dv_0 = \frac{1}{\sigma_v \sqrt{2 \pi}} \exp\left[- \frac{(v_0 - \overline{v_0})^2}{2 \sigma_v^2}\right]$$\
$$\times \frac{1}{\sigma_{\gamma} \sqrt{2 \pi}} \exp\left[- \frac{(\gamma - 1)^2}{2 \sigma_{\gamma}^2}\right] d\gamma \, dv_0.$$ In Fig. 1 we show various contours corresponding to different value of $P (\gamma, v_0)$ (excluding the constant pre-factor). The probability that $\gamma$ and $v_0$ at the end of a cycle jointly lie within a certain area in Fig. 1 is easily obtained by integrating $\int P (\gamma, v_0) \, d\gamma \, dv_0$ over that area. On carrying out this integration over the shaded region in Fig. 1 that corresponds to conditions for producing grand minima, we find the value of the integral to be $0.013$ (i.e., $1.3\%$), remarkably close to the probability of occurrence of grand minima on the basis of observational data @usos07. Table 1 gives the value of this integral for a few combinations of $\sigma_v$ and $\sigma_{\gamma}$. It may be noted that there would be contributions to the probability from the regions to the left and to the bottom of the shaded region in Fig. 1. We have checked that these contributions are negligible.

<span>**Table 1.**</span> <span>This table gives the values of the probability (in %) of the initiation of grand minima (given by $\int P (\gamma, v_0) \, d\gamma \, dv_0$ over the shaded region of Fig. 1) for different combinations of the standard deviations $\sigma_v$ and $\sigma_{\gamma}$.</span>

  ------------ ------ ------ ------------------- ------
                              $\sigma_{\gamma}$  
                       0.25         0.35          0.46
                3.20   0.8           1.3          1.7
   $\sigma_v$   3.34   0.9           1.3          1.9
                3.50   1.1           1.7          2.2
  ------------ ------ ------ ------------------- ------

![image](fig3.eps){width="90.00000%"}

To check whether grand minima really do occur in accordance with the above simple probability estimate, we have carried out extensive simulations on the basis of our dynamo code @nc02 [@cnc04]. Karak @k10 changed the values of a few parameters and our present simulations are based on this model. We introduce fluctuations in  generation by the method proposed by Choudhuri et al. @ccj07. At the end of every cycle, we multiply the  above $0.8 R_{\odot}$ by the random number $\gamma$ obeying the Gaussian distribution shown in Fig. 2b. This procedure introduces fluctuations in the poloidal field generated in the last cycle lying in the upper portions of the convection zone, whereas any poloidal field produced at the earlier cycles lying at the bottom of the convection zone remains unchanged. While this procedure introduces a momentary discontinuity in the field lines at depth $0.8 R_{\odot}$ (see Fig. 1 in Jiang et al. @ccj07), this discontinuity disappears soon and does not cause any problem. To introduce fluctuations in , we change its amplitude randomly after every 30 yr such that the amplitudes obey the Gaussian distribution shown in Fig. 2a. On running the code for 11,000 yr in this way, we obtain the number of grand minima in the range 24–30 for different realizations of randomly generated  and $\gamma$—remarkably close to the observational finding of 27 grand minima in the last 11,000 yr. Another important result is that we find the Sun to spend about 10–15% of the time in a grand minimum state, which is very close to 17% found in the observational study @usos07. Fig. 3 shows the durations of grand minima and their times of occurrence for a particular run. We ourselves have been amazed that the observational data are reproduced so well on running our code with the simple incorporation of the fluctuations suggested by the histograms in Fig. 2, without having to change any parameters of the dynamo model compared to our previous work. Fig. 4 is a sample plot showing how the sunspot number varied for a typical 1000 years during which two grand minima occurred. Histograms of the duration of grand minima and waiting time between grand minima are shown in Fig. 5. These histograms are to be compared with Figs. 7 and $6$ in Usoskin et al. @usos07. In the simulations presented above, we have changed meridional circulation abruptly after every 30 yr. To check how the above results change with this coherence time $\tau$, we have done several simulations with different $\tau$. We see that even when $\tau \sim 15$ yr, we get around 15–20 grand minima. However, when $\tau$ is less than this value, the number of grand minima gets very much reduced. Another important point: instead of changing  abruptly at a time, if we change it smoothly in few years, then also the results do not change significantly.

![The plot of sunspot number against time for a typical 1000 years. The two grand minima around 1100 and 1500 years can be seen in Fig. 3 as well.<span data-label="simu_grand"></span>](fig4.eps){width="60.00000%"}

![Left panel shows the distribution of the durations of the grand minima and the right panel shows the distribution of the waiting times between the grand minima.<span data-label="simu_grand"></span>](fig5.eps){width="60.00000%"}

When sunspots are absent, the Babcock–Leighton process for the generation of  cannot take place. Presumably, during a grand minimum, the  has to be generated by the $\alpha$-effect originally proposed by Parker @parker55 and Steenbeck, Krause & Rädler @steen. It is possible that an $\alpha$-effect coexists along with the Babcock–Leighton mechanism all the time, although its nature and spatial distribution (even its sign) is completely unclear at this time. In view of this uncertainty, our results presented in Figs. 3 and 4 are obtained by using the same form of $\alpha$ at all times. Once the Sun is pushed into a grand minimum, it comes out of the grand minimum in a time of the order of dynamo growth time, as discussed in detail by Choudhuri & Karak @ck09. One intriguing fact to note is that we get durations of grand minima similar to their observed values when we keep using the same $\alpha$ throughout the grand minima.

We conclude that the irregularities of solar cycles including the grand minima are produced by fluctuations in poloidal field generation and in meridional circulation. Assuming these fluctuations to obey Gaussian distributions, we obtain the basic parameters of the Gaussians from the distribution of the peak sunspot numbers and the durations of the last 28 solar cycles. On running our code with such fluctuations, we find that our theoretical dynamo model produces grand minima at a frequency remarkably close to what is found in the data for the last 11,000 yr.

This work is partially supported by JC Bose fellowship of DST awarded to ARC. BBK thanks CSIR, India for financial support. ARC is grateful to Saku Tsuneta for the warm hospitality in NAOJ where the revised version of the paper was produced.

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{'source': 'arXiv_out/1208'}