Shape of the 11-Year Cycle of Solar Activity and the Evolution of Latitude Characteristics of the Sunspot Distribution

Abstract

The solar activity index and parameters of the spatial distribution of sunspots are known to be related. Using these relationships, we propose interrelated approximations for the sunspot number (SN) and the two key latitude characteristics of their distribution in the cycle: the mean latitude of sunspots and the standard deviation of their latitudes. The two parameters of these approximations are the cycle amplitude SN_max and the drift of its downward branch relative to the cycle beginning t₀. These approximations specifically take into account the relationship between amplitudinal and spatial properties of the 11-year solar cycle, as well as the universality of the behavior of the activity and mean latitude of sunspots in the declining phase of the cycle. We demonstrate that the pair of parameters SN_max and t₀ allows approximation of both the shape of the cyclic curve and the latitude–time diagram for sunspots of this cycle (“Maunder’s butterfly”).

1 INTRODUCTION

The search for dependences describing the behavior of the solar activity index in the 11-year solar cycle dates back to the 19th century. Rudolf Wolff tried to describe a dependence by superimposing four sinusoids corresponding to the phases of the motion of the solar system planets (see Waldmeier, 1935); over the next 150 years, there had been multiple attempts to find an appropriate dependence (see, e.g., Vitinskii et al. (1986), Hathaway et al. (1994), and references therein).

These approximations often have a formal mathematical character: finding a function with a small number of parameters that is most consistent with the behavior of the given index. However, in describing the approximating curve, it is useful to take into account the relationships between sunspot distribution features.

It is known that the evolution of solar activity during the 11-year cycle can be divided into two qualitatively different phases. The inclining and declining phases are characterized by a comparatively rapid increase and a slower decrease in activity, respectively. The activity in the inclining phase increases more quickly for higher cycles (“the Waldmeier rule”); on the contrary, in the declining phase its behavior depends weakly on the prehistory of this cycle, but it is well correlated with the current mean latitude of sunspots (Eigenson et al., 1948; Gnevyshev and Gnevysheva, 1949; Hathaway, 2011; Ivanov and Miletsky, 2014). The latter is useful because it allows one to construct two interrelated approximating curves: the first describes the evolution of the activity index in the solar cycle, and the second describes the behavior of the average latitude of sunspots.

Cameron and Schüssler (2016) showed that the specific behavior of solar activity in the declining phase of the cycle may be related to the fact that the decrease of the magnetic field strength at this time happens predominantly due to magnetic diffusion, and they proposed a quantitative model of this mode of activity.

In this study, we propose two consistent approximating functions that describe both the behavior of the activity index in the 11-year cycle and the latitudinal distribution of sunspots in this cycle; the form of these sunspots takes into account the aforementioned relationships and regularities.

2 DATA

The data used by us are the annual-mean values of the recalibrated sunspot number SN for 1700–2016 (WDC-SILSO, Royal Observatory of Belgium, Brussels, http://www.sidc.be/silso/DATA/SN_y_tot_V2.0.txt). Our calculations of the spatial characteristics of sunspots for 1874–2016 are based on the Greenwich sunspot catalog (Greenwich – USAF/NOAA Sunspot Data, http://solarscience.msfc.nasa.gov/greenwch.shtml); the average latitude of sunspot groups φ and the standard deviations σ_φ of these latitudes (regardless of their sign) for the given year are calculated for these data. The time intervals used below correspond to the Greenwich catalog (1874–2016; i.e., cycles 11(12)–23) unless otherwise noted.

3 RELATIONSHIP BETWEEN SN, φ, AND σ_φ

Figure 1 shows the correlation between the annual-mean SN values and the mean latitude of sunspots φ. In the declining cycle phase (black circles), there is a dependence between these indices (Ivanov and Miletsky, 2014) with a correlation coefficient of R = 0.93, which can be described by the linear regression

$$\varphi \left( {{\text{SN}}} \right)~ = {{a}_{\varphi }} + {{b}_{\varphi }}{\text{SN}} = 8.37^\circ + 0.0338^\circ {\text{SN}}.$$

((1))

Fig. 1.

Relationship between the annual-mean sunspot latitude φ and the SN index for 1874–2016. The black circles indicate the years of the declining phase, and the dotted line stands for regression (1).

The standard deviations of latitudes σ_φ are also related to the SN (Ivanov et al., 2011) (Fig. 2a). If the years of activity minima and their adjacent years (indicated by empty circles in the figure), when the old and new 11-year cycles often overlap and it is difficult to determine their parameters unambiguously, are disregarded, this dependence can be described by the regression

$$\begin{gathered} {{\sigma }_{\varphi }}\left( {{\text{SN}}} \right) = {{a}_{\sigma }} + {{b}_{\sigma }}{\text{SN}} = 4.01^\circ + 0.0150^\circ {\text{SN}} \\ \left( {R = 0.87} \right). \\ \end{gathered} $$

Fig. 2.

Relationship between the annual-mean activity parameters for 1874–2016: (a) (SN, σ_φ), the dotted line by regression (2) and (b) (φ, σ_φ), the dotted line by regression (3). The regressions in both diagrams were constructed for years that are more than one year away from cycle minima (black circles).

Obviously, one can also construct a third linear regression describing the correlation between σ_φ and φ in the declining cycle phase (Fig. 2b):

$${{\sigma }_{\varphi }}\left( \varphi \right) = 0.459\varphi --~0.0401^\circ {\text{ }}\left( {R = 0.91} \right).$$

((3a))

Since the free term of this linear regression is small, it can be dropped without loss of accuracy:

$${{\sigma }_{\varphi }}\left( \varphi \right) = p\varphi = 0.456\varphi .$$

((3b))

4 SN APPROXIMATION

Cameron and Schüssler (2016) proposed a mechanism describing the behavior of activity in the declining phase. Assuming that the intensity of the toroidal magnetic field in this phase decreases mainly due to diffusion through the equator and that the latitudinal field profiles can be approximately described by normal distributions, these authors obtained for description of the SN a function F that satisfies the equation

$$\frac{{dF}}{{dt}} = - C\frac{\varphi }{{\sigma _{\varphi }^{3}}}F.$$

((4))

Here,

$$C = {{\left( {\frac{{180^\circ }}{\pi }} \right)}^{2}}\sqrt {\frac{2}{\pi }} {{\left( {\frac{\eta }{{{{R}^{2}}}}} \right)}_{{{\text{eff}}}}}\exp \left( { - \frac{{{{\varphi }^{2}}}}{{2\sigma _{\varphi }^{2}}}} \right),$$

where η is the coefficient of turbulent magnetic diffusion, R² is the characteristic square of the radius of the magnetic field generation region, and ${{\left( {\frac{\eta }{{{{R}^{2}}}}} \right)}_{{{\text{eff}}}}}$ is the effective ratio of the coefficients in this region. With relation (3b), the exponent and, hence, C are constants. Then, Eq. (4) is a differential equation of the first order, which, in view of (2) and (3), can be written as

$$\frac{{dF}}{{dt}} = - \frac{A}{{{{{(F + \alpha )}}^{2}}}}F,$$

((5))

where $A = \frac{C}{{pb_{\sigma }^{2}}}$, $\alpha = \frac{{{{a}_{\sigma }}}}{{{{b}_{\sigma }}}}$, and its solution is

$${{F}_{2}}(t;{{t}_{0}}) = f(A({{t}_{0}} - t)),$$

((6))

where f = g^–1 is the inverse to the function

$$g(y) = {{\alpha }^{2}}\log y + 2\alpha y + {{y}^{2}}/2.$$

Thus, the activity in the declining phase can be described by curve (6), which has the same form for all cycles for fixed A and α, and the parameter t₀ determines the shift of the declining branch of this cycle along the time axis (hereafter, time t is assumed to be counted from the cycle minimum). It was noted earlier that it is such a universal behavior of the activity index (unrelated to the cycle intensity) that is typical for the declining cycle phase. In this case, the parameter α = 248 is equal to the ratio of the coefficients of linear regression (2), and the second empirical parameter A = 4.17 × 10⁴ year^–1, which determines the extension of the standard curve f(x) along the ordinate axis, was found by minimizing the sum of standard deviations between the observed index and solution (6) in the cycle declining phases for the entire SN data series (1700–2016).

Unlike the cycle declining phase, the behavior of activity in the inclining phase depends strongly on cycle intensity (according to the Waldmeier rule). The approximating function for this phase must have a minimum at t = 0, sufficiently rapid growth, and a maximum (SN_max) at some moment t = T_max. With no aim of describing in detail the behavior of activity in the inclining phase, we choose the simplest function of this form

$${{F}_{1}}(t;{\text{S}}{{{\text{N}}}_{{\max }}},{{T}_{{\max }}}) = {\text{S}}{{{\text{N}}}_{{\max }}}{{\sin }^{2}}\frac{{\pi t}}{{2{{T}_{{\max }}}}}.$$

((7))

The function approximating SN behavior in the full cycle is proposed to be a piecewise-defined function F taking values of F₁ and F₂ at the time intervals 0 ≤ t ≤ t* and t ≥ t*, respectively. Here, the boundary point t* is chosen such that F is continuous and differentiable on the domain of definition. This can be achieved by requiring that the corresponding curves be tangent to one another at t*:

$$\begin{gathered} {{F}_{1}}(t{\text{*}}) = {{F}_{2}}(t{\text{*}}), \hfill \\ F_{2}^{'}(t{\text{*}}) = F_{2}^{'}(t{\text{*}}). \hfill \\ \end{gathered} $$

It is rather clear (Fig. 3a) that, if SN_max < F₂(0), the point t* exists and is unique on the interval from T_max to 2T_max. We do not strictly prove this fact; for our purposes, it is sufficient that this point can be found for all cycles of the full SN data series (see Table 1). The condition imposed on F₁ unambiguously relates the parameters t₀, SN_max, and T_max. We have the freedom of choice: we can choose t₀ and SN_max or t₀ and T_max as a pair of parameters specifying the function F. We choose the first option: SN_max is assumed to be equal to the observed cycle amplitude and the length T_max is uniquely expressed in terms of t₀ and SN_max (thus, it does not have to be equal to the observed length of the inclining phase T_max,obs, though it is close to it).

Fig. 3.

(a) Search for the conjugation point of the curves of F₁ and F₂. The curves indicated by solid lines and marked with the numbers from 1 to 5 correspond to functions F₁(t; SN_max, T_max) with a fixed SN_max = 240 and successively increasing T_max; curve F₁ number 3 is tangent to the curve F₂(t; t₀) at the point with abscissa t*. (b) Solid lines are the approximating curves F(t; t₀, SN_max) corresponding to the cycle amplitudes SN_max = 80, 120, 160, 200, 240. The dotted line on both graphs denotes the function F₂(t; t₀) for t₀ = 15 years. In the graph (b), this function describes the behavior of activity common for all SN_max in the cycle declining phase.

Table 1.   Characteristics of functions F(t; t₀, SN_max) approximating the SN index in individual solar cycles: t₀ and SN_max are the function parameters, T_max is its time of maximum, t* is the boundary of intervals, Δ is the rms error of the approximation

Thus, we have described the class of piecewise-defined smooth functions F(t; t₀, SN_max) with two parameters (see Fig. 3b) describing the behavior of the 11-year cycle in both phases. Table 1 gives the approximation parameters for the cycles of the full SN data series, as well as the rms approximation errors Δ. Figure 4 shows the approximating curves for the Greenwich epoch.

Fig. 4.

Annual-mean SN (top) and annual-mean sunspot latitude φ (bottom). The circles correspond to observed values and the lines correspond to the approximations F(t;t₀, SN_max) and F_φ(t; τ(t₀)), respectively.

In terms of approximation accuracy, the proposed functions may compete well with other well-known two-parameter formulas. For example, when the SN is approximated by functions of the Stewart–Panofsky type S₃(t; C, β) = Ct³exp(–βt) (Steward and Panofsky (1938); see also Vitinskii et al. (1986)) the rms error on the full SN data series is 14.6, and for the functions chosen by us it is 17.8. Here, our approximation takes precedence over S₃, because the form of decreasing branches of functions F is consistent with the universal law of activity decrease in the declining phase (Fig. 3b).

Although the functions F(t; t₀, SN_max) depend upon two parameters, values of these parameters corresponding to the full SN data series are not completely independent. In this case, the correlation between SN_max and t₀ is R = 0.49, increasing up to R = 0.73 if only the Greenwich cycles starting with the 12th cycle are taken into account (see Fig. 5). The Greenwich epoch is characterized by the following linear relationship between the parameters of the functions:

$${{t}_{0}}\left( {{\text{S}}{{{\text{N}}}_{{{\text{max}}}}}} \right) = 13.2 + 0.00862{\text{ S}}{{{\text{N}}}_{{{\text{max}}}}}\left( {R = 0.73} \right).$$

((8))

Fig. 5.

Relationship between the approximation parameters t₀ and SN_max. The black circles indicate the cycles of the Greenwich epoch (starting from the 12th cycle), and the light circles indicate the cycles of the pre-Greenwich epoch. The dotted line is regression (8).

In the pre-Greenwich epoch, the corresponding correlation decreases to R = 0.40, which is caused, probably, by increased error in determining of the SN in this epoch.

It should be noted that the approximation chosen by us leads naturally to the Waldmeier rule. As mentioned above, the length of the declining phase of the approximating curve T_max is functionally associated with t₀ and SN_max. If the shift in t₀ were the same for all cycles, T_max would decrease with SN_max growth (see Fig. 3b) and the correlation between these parameters would be close to –1. However, in real data, t₀ increases with cycle amplitude according to (8); here, the curve of F₂ shifts rightward and T_max increases. Thus, there is no full anticorrelation between SN_max and T_max. Nevertheless, the correlation between these parameters remains sufficiently strong (Fig. 6):

$${{T}_{{{\text{max}}}}}\left( {{\text{S}}{{{\text{N}}}_{{{\text{max}}}}}} \right) = 6.56\,~ - \,0.0129{\text{ S}}{{{\text{N}}}_{{{\text{max}}}}}\left( {R = -0.78} \right).$$

((9))

Fig. 6.

Relationship between the cycle amplitude SN_max and the growth phase length T_max for approximating functions to illustrate the Waldmeier rule. The dotted line corresponds to regression (9).

5 APPROXIMATION OF THE CURVE OF MEAN LATITUDES OF SUNSPOTS

It is well known that the mean sunspot latitudes φ decrease monotonically with the evolution of the 11-year cycle (“Spörer’s law”), and it has repeatedly been shown that their behavior depends weakly on the cycle amplitude and can be described, for example, by an exponential dependence (Hathaway, 2011; Roshchina and Sarychev, 2011; Ivanov and Miletsky, 2014):

$${{F}_{\varphi }}(t;\tau ) = {{\varphi }_{0}}\exp \left( {\frac{{\tau - t}}{\beta }} \right),$$

((10))

where φ₀ = 15° and β = 8.3 years are empirical coefficients common for all cycles and τ is a cycle-dependent parameter that determines the shift of the exponent along the time axis. Table 2 shows the parameters τ and the rms approximation errors Δφ for the Greenwich epoch cycles.

Table 2.   Characteristics of functions F_φ(t; τ) approximating the mean sunspot latitude in individual solar cycles: τ is the function parameter and Δ_φ is the rms error of the approximation

It follows from relation (1) that τ must be related to the cyclic curve parameters. Indeed, the following regression can be obtained for τ:

$$\tau \left( {{{t}_{0}}} \right) = 0.61~{{t}_{0}}-4.30{\text{ }}\left( {R = 0.80} \right).$$

((11))

Thus, the two parameters t₀ and SN_max can be used to describe both the level of cycle activity and one of the characteristics of the spatial distribution of sunspots. Approximations of the mean latitude in cycles corresponding to dependence (10) with the parameter determined from (11) are shown on the bottom panel of Fig. 4.

6 SYNTHESIS OF THE BUTTERFLY DIAGRAM

In a good approximation, the distribution of sunspots by latitude is normal (Ivanov et al., 2011) and the variance of this distribution $\sigma _{\varphi }^{2}$ is determined by relation (2); therefore, the approximations obtained above for the SN and φ can also be used to describe the distribution of sunspots by latitude. This means that, knowing the pair of parameters that determine the behavior of the activity level (t₀ and SN_max), one can describe in some approximation their spatial distribution as well.

Figure 7 shows two time–latitude diagrams for sunspots (“Maunder’s butterflies”). The top diagram was obtained from the observed data, and the bottom diagram was constructed with the use of parameters t₀ and SN_max (see Table 1). Here, for each year and hemisphere, we generated a random sequence of sunspot groups, the number of which corresponded to half of the full SN index calculated with the help of the function F and the moments of appearance were distributed uniformly over the year, and the latitudes were distributed according to the normal law with a mean latitude φ and its variance σ_φ obtained from approximation (10) and regressions (11) and (2).

Fig. 7.

The observed butterfly diagram (top) and its synthesized version constructed from approximating functions (bottom). For clarity, only one of the ten sunspot groups is shown on each diagram.

It can be seen that the observed and synthesized “butterfly diagrams” are quite consistent in their form. Some differences between the diagrams are explained by the facts that (a) the actual distribution of sunspots is slightly different from the normal distribution (Ivanov et al., 2011) and (b) the mean sunspot latitude at minima is not described by an approximating function (10) (see Fig. 4) because of its strong dependence on the details of the distribution of sunspots in the old and new cycles.

Since the calculation of approximation parameters requires only data for the activity level, we can also construct similar diagrams for the pre-Greenwich epoch, for which there are no systematic data on sunspot latitudes (Fig. 8).

Fig. 8.

The synthesized butterfly diagram for the pre-Greenwich epoch. For clarity, only one of the ten sunspot groups is shown on the diagram.

7 CONCLUSIONS

Thus, we have constructed a class of functions F(t; t₀, SN_max) that approximate the SN behavior and depend on two parameters of the cycle: its height SN_max and the shift of the activity curve in the declining phase t₀. The descending branches of these functions have a universal form and differ only by the shift along the time axis. This feature agrees with the observed form of the 11-year cycle of solar activity and can be caused by the special diffusion mode of toroidal magnetic fields in the declining phase of the 11-year solar cycle.

The fact that the activity in the declining phase correlates well with the mean sunspot latitude allowed us to construct another class of approximating functions F_φ(t; τ), describing the behavior of the mean latitudes. These functions depend on a single parameter τ, which can be related to t₀ by relation (11).

Finally, the two parameters of the approximation proposed by us turn out to be sufficient to describe both the form of the cyclic curve (i.e., the activity level behavior) and resulting evolution of characteristics of the latitudinal sunspot distribution (and, as a result, the “butterfly diagram” for the given cycle).