Origin of Grand Minima in Sunspot Cycles

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 1645–1715. Analyses of cosmogenic isotopes (${}^{14}\mathrm{C}$ and ${}^{10}\mathrm{Be}$) indicated that there were about 27 grand minima in the last 11 000 yrs, implying that about $2.7\%$ of the solar cycles had conditions appropriate for forcing the Sun into grand minima. We address the question of 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 the last 28 solar cycles, we show from our flux transport dynamo model that about $1–4\%$ of the sunspot cycles may have conditions suitable for inducing grand minima.

Figure 1

The solid line shows the values of the meridional circulation amplitude $v_0$ and the poloidal field scale factor $\gamma$, which produce grand minima of duration $\sim 20\mathrm{yrs}$. 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 prefactor) given in the plot. The scale factor $\gamma$ is defined as the amplitude of the 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 are given by Karak [21]).

Figure 2

(a) Histogram of the meridional circulation amplitude derived from the observed periods of the last 28 solar cycles. The solid curve is the Gaussian function with $\mathrm{\text{mean}}=23\mathrm{m}{\mathrm{s}}^{-1}$ and standard deviation ${\sigma }_v=3.34\mathrm{m}{\mathrm{s}}^{-1}$. (b) Histogram of the poloidal field scale factor derived from the peak sunspot number of the last 28 solar cycles. The solid curve is the Gaussian function with $\mathrm{\text{mean}}=1$ 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.

Figure 3

The durations of grand minima indicated by vertical bars at their times of occurrence in a 11 000 yr simulation. This is the result of a particular realization of random fluctuations that produced 28 grand minima.

Figure 4

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.

Figure 5

The 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.