Solar-cycle-related Variation of Differential Rotation of the Chromosphere

More than 35,000 daily observed spectroheliogram images were taken in the chromospheric resonance line of Ca ii K at the Mount Wilson Observatory during the period of 1915 August 10 to 1985 July 7 (Foukal & Milano 2001; Lefebvre et al. 2005; Ermolli et al. 2009; Bertello et al. 2010; Pevtsov et al. 2016; Bertello et al. 2020). These images have been digitized, calibrated, selected, and transformed into corresponding Carrington rotation (CR) synoptic maps (Bertello et al. 2010; Sheeley et al. 2011; Bertello et al. 2014). CR synoptic maps are very beneficial when studying the dynamic behavior of solar activity and how it changes at each location on the solar surface (Chowdhury et al. 2022). Nine hundred and thirty-eight CR synoptic maps of Ca ii K normalized intensity from CR827 to CR1764 were obtained, with some data gaps from CR1147 to CR1150 (Bertello et al. 2020). They are available from the website https://doi.org/10.7910/DVN/RQSEJ8. In each CR synoptic map, longitude is divided into 360 equidistant values, and the sine of latitude is also divided into 180 equidistant values from −1 (the southern pole) to 1 (the northern pole) (Bertello et al. 2014, 2020).

In Figure 1, we present a CR synoptic map in this database to characterize the Ca ii K intensity features. As the figure displays, brightening is evident at low latitudes (± 40°). Figure 2 shows the temporal–latitudinal distribution of Ca ii K normalized intensity during the period of 1915 August 10 to 1985 July 7, spanning seven solar cycles (from cycle 15 to cycle 21), and thus there are seven butterfly patterns that can be seen at low latitudes (± 40°) and which have intensity values that are distinctly larger than those of the background. These butterfly patterns represent the active chromosphere, lying over the strong magnetic field of the photosphere. In the following research, we will only study the chromospheric rotation in the low-latitude zone (± 40°).

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To study the characteristics of long-term variations in the chromospheric rotation and solar activity, we take advantage of the yearly mean sunspot numbers (version 2) from the Solar Influences Data Analysis Center (SIDC), which is the most commonly used measure of long-term solar activity. They can be obtained via http://sidc.oma.be. Further, to compare the north–south (N–S) asymmetry of the chromospheric rotation rate with that of solar activity, we made use of the Greenwich daily sunspot area time series of the opposite hemispheres for the same time period. They can be downloaded freely from the website http://solarcyclescience.com/activeregions.html.

As we mentioned above, there are some data gaps from CR1147 to CR1150. To ensure the Ca ii K line time series are even, these data are filled with a large negative value. To acquire the yearly rotation rate, we use the classical autocorrelation analysis method to determine the rotation period (Stimets & Londono 1982; Xu & Gao 2016; Ray et al. 2018; Li et al. 2019, 2020). Autocorrelation is the linear dependence of a variable with itself at two points in time (Box et al. 2010). The autocorrelation coefficient at lag k is given by

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{{\boldsymbol{k}}}=\displaystyle \frac{{\sum }_{i=1}^{N-k}\left({x}_{i}-\mu \right)\left({x}_{i+k}-\mu \right)}{{\sum }_{i=1}^{N}{\left({x}_{i}-\mu \right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where x_i is the time series of length N and μ is its overall mean, $\mu =\displaystyle \frac{1}{N}\sum _{i\,=\,1}^{N}{x}_{i}$. When calculating the autocorrelation coefficient (CC), if a negative value is encountered, both it and its paired value are excluded. Therefore, the calculation of the CC is limited to available values, without applying any negative value or interpolation. For a time series of a certain latitude in a certain year, the calculated CC varies with the relative phase shift of the series relative to itself, and the rotation period is the shift value corresponding to the local maximum CC around 27 days. As an example, Figure 3 displays the calculation results of CC for a certain year obtained at four different latitudes. The Ca ii K line time series during the period of 1915 August to 1985 July contains a total of 71 yr, and we get 118 rotation periods at 118 measured latitudes within the low-latitude zone (± 40°) in each year. It should be noted that the period obtained in Figure 3 is the Carrington synodic rotation period (CR), which is defined as 27.275 days, and its corresponding rotation rate is 360° /(27.275 days), i.e., 13199/day. After the synodic rotation period (P) of a series is obtained, the rotation rate (R) of a certain latitude in a certain year can be calculated by this formula: R = P × 13.199 (in °/day, (Shi & Xie 2013; Lamb 2017; Li et al. 2019; Deng et al. 2020; Li et al. 2020)). Ultimately, we acquire 71 yearly rotation rates at each of the 118 measured latitudes.

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Variation of differential rotation rate (ω(ϕ)) with latitude at middle- and low-latitude zones is traditionally expressed by the standard formula (Newton & Nunn 1951)

$$\begin{eqnarray}&&\omega (\phi )=A+B{\sin }^{2}\phi ,\end{eqnarray} \tag{ 2 }$$

where ω(ϕ) is the solar sidereal angular velocity at latitude ϕ, the parameter A represents the equatorial rotation rate, and the parameter B is the latitudinal gradient of rotation (Howard 1984). We make use of this formula to fit the yearly rotation rate of the Ca ii K line time series that we have obtained via the above autocorrelation method. Figure 4 shows the cross-correlation coefficient of the formula fitting to the latitudinal distribution of the yearly rotation rate. From the figure, we can see that all calculated correlation coefficients are larger than the critical value for the correlation coefficient at a 95% confidence level, except for the year of 1933, which means the formula can statistically fit the yearly rotation rate well. The fitting values of A and B for the year of 1933 are substituted by linear interpolation of their adjacent two points. It is very clear that the correlation is higher in the maximum year of a solar cycle than the corresponding minimum year.

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Variation of the parameters A and B within solar cycles is shown in Figure 5. The parameter A is found to present a long-term weak downward trend, but the correlation coefficient from a linear fitting is −0.027, without statistical significance. The average value of A over all solar cycle maxima (13.496 (± 0.084) deg/day) is found to be slightly larger than that of all solar cycle minima (13.450 (± 0.073) deg/day). Also, a linear fitting to the parameter B over time t (years) yields B = 9.0916 − 0.0057 × t, with a correlation coefficient of 0.244, which is statistically significant at the 95% confidence level, thus the parameter B shows a relatively obvious decreasing trend. The average value of B over all solar cycle maxima (−2.468 (± 0.656) deg/day) is found to be slightly smaller than that over all solar cycle minima (−2.140 (± 0.883) deg/day).

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We display the variation of the yearly mean sunspot numbers within solar cycles in the bottom panel of Figure 5. The linear fitting of the yearly mean sunspot numbers over time shows a relatively slight increasing trend. By comparing the relationship of each of parameters A and B to the yearly mean sunspot numbers, we find that there is a significant positive correlation relationship between parameter A and the yearly mean sunspot numbers, with a correlation coefficient of 0.347. However, parameter B has a significant negative correlation with the yearly mean sunspot numbers, with a correlation coefficient of −0.388. That is, the magnetic field is stronger and the parameter B is smaller, clearly indicating that there seems to be a correlation between field strength and chromospheric differential rotation, in sharp contrast to what happens in the photosphere. In the photosphere, the strong magnetic field should suppress differential rotation (Brajša et al. 2006; Wohl et al. 2010; Li et al. 2013a, 2013b).

According to the obtained parameters A and B, we utilize the standard formula to calculate the latitudinal distribution of the yearly rotation rate within a ± 40° latitude belt in the considered time frame, as shown in Figure 6. The rotation rate has a distinct characteristic of migration during a solar cycle. Figure 7 displays four isopleth lines of rotation rates, and their corresponding rotation rates are 13.3, 13.1, 12.9, and 12.7 (deg/day) from low to high latitudes. Combining Figures 6 and 7, we find that the migration of rotation rates seems to be hindered in the descending phase of a solar cycle, which is a dramatic contrast to the oscillation pattern of solar surface differential rotation (Snodgrass 1987; Li et al. 2013b).

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To study the change of parameter B over different phases of a solar cycle, we calculate the average value of B for all considered time frame within the same solar cycle phase relative to the respective minimum/maximum years of the nearest solar cycle, and its corresponding standard errors. The results are shown in Figure 8. As the figure displays, the absolute value of B increases continuously in the first three years of a solar cycle, peaks around the maximum time, then decreases slowly from the third year to the eighth year, and finally slowly increases again from the eighth year to the end of the solar cycle, clearly indicating a negative correlation between the parameter B and solar magnetic field. Therefore, there seems to be a correlation between field strength and chromospheric differential rotation. Such a solar-cycle phase of the parameter B in the chromosphere is clearly different from that in the photosphere, indicating that the effect of magnetic fields on differential rotation is different in the two atmosphere layers. In the photosphere, parameter B is weakly correlated to solar cycles and changes little during the ascending phase of a solar cycle. The absolute value of B is large, but the rotation rate on average is generally small. Therefore, the negative correlation implies that, on the whole, the stronger magnetic field is, the slower rotation rate is. As Figure 7 shows, at a certain latitude, the rotation rate is generally smaller around the maximum time of a solar cycle than in the rest time of the solar cycle.

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Many kinds of solar activities appear unevenly in the two hemispheres. This feature is known as the hemispheric north–south (N–S) asymmetry, which was first found by Spoerer (1889) through the long-term observations of sunspots. A number of authors have found the N–S asymmetry to exist in solar differential rotation (Gilman & Howard 1984; Antonucci et al. 1990; Hathaway & Wilson 1990; Javaraiah & Gokhale 1997; Brajša et al. 2000; Gigolashvili et al. 2005; Vats & Chandra 2011; Li et al. 2013a; Zhang et al. 2015; Bagashvili et al. 2017; Xie et al. 2018; Sharma et al. 2020; Hrazdíra et al. 2021; Edwards et al. 2022). Therefore, it is important to study the temporal variation of the N–S asymmetry of solar differential rotation rates, not only to provide clues to better understand the underlying mechanisms of solar cycles but also to study the N–S asymmetry of solar activities. Next, we will investigate the N–S asymmetry of the chromospheric rotation rate and its relation to the N–S asymmetry of solar activity.

To more clearly represent the rotational asymmetry, we utilize the same second-order polynomial that is used by Li et al. (2013a) to fit the latitudinal distribution of the chromospheric rotation rate:

$$\begin{eqnarray}&&\omega (\phi )={C}_{0}+{C}_{1}\sin \phi +{C}_{2}{\sin }^{2}\phi ,\end{eqnarray} \tag{ 3 }$$

where C₀, C₁, and C₂ are the polynomial coefficients, ϕ is latitude in the southern hemisphere (negative values) and the northern hemisphere (positive values), ω(ϕ) is the rotation rate at a latitude ϕ, and C₁ is here defined as the "asymmetry coefficient." When C₁ is positive (negative), it means that the northern (southern) hemisphere should rotate faster than the southern (northern) hemisphere, and that the differential of rotation rate is more obvious in the southern (northern) hemisphere than that in the northern (southern), correspondingly.

The result obtained is shown in the top panel of Figure 9. During cycles 15, 16, 19, 20, and 21, the southern hemisphere rotates faster, whereas in cycles 17 and 18, the northern hemisphere rotates faster. The degree of asymmetry during the minima of solar cycles is generally higher than that during the maxima of the solar cycle. Furthermore, to understand the evolution trend of the "asymmetry coefficient" in the considered time frame, the linear fitting method is used to fit the asymmetry coefficient, and the result presents a long-term downward trend. The correlation coefficient of 0.197 is below the 95% confidence level.

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To study the relationship between the N–S asymmetry of the chromospheric rotation and that of solar activity, we calculate the yearly N–S asymmetry value of the Greenwich daily sunspot area time series of the opposite hemispheres for the same time period. The N–S asymmetry is defined by the normalized asymmetry index (NAI), which was first proposed by Newton & Milsom (1955): N_asy= (N − S)/(N + S), where N and S stand for the yearly numbers of sunspot areas in the northern and southern hemispheres, respectively. The obtained values are plotted in the bottom of Figure 9. In fact, Li et al. (2002) have determined the dominant hemisphere of sunspot area in solar cycles 12–23: they found that, during cycles 15, 16, 19, and 20, the northern hemisphere dominates, whereas for the rest of the cycles, such as 17, 18, and 21, it is uncertain which hemisphere is dominant. The N–S asymmetry of sunspot area is found to present a long-term weak downward trend in the considered time frame by the linear fitting, but the correlation coefficient is 0.135, below the 95% confidence level.

Comparing the two panels of Figure 9, we find that there exists a significant negative correlation between the N–S asymmetry of the chromospheric rotation rate and that of solar activity over the considered time frame, and their correlation coefficient is −0.337, which is statistically significant at the 95% confidence level. Such a negative correlation is significantly reflected in the fact that C₁ is a negative value when asymmetry of sunspot activities is a large positive value. This indicates that the differential of the rotation rate is more obvious in the northern hemisphere than in the southern, and that on the whole, the rotation rate is greater in the southern hemisphere than that in the northern, when the magnetic field strength is clearly larger in the northern hemisphere than that in the southern. This implies that there seems to be a correlation between field strength and chromospheric differential rotation.