In this paper, by applying the wavelet transformation analysis to the data of the daily 10.7 cm radio flux covering the period from 1947 February 14 to 2014 August 31, a significant period of about 27 days can be found, indicating the existence of rotational modulation in the temporal variation of the daily 10.7 cm radio flux. Then, the solar coronal rotation periods are obtained based on the result of the wavelet transformation analysis, and the temporal variation of the coronal rotation is revisited. We find that there exist significant periods of about 2.1, 3.0, 4.5, 6.6, 8.6, and 10.3 yr in the temporal variation of the coronal rotation. A possible period of 22.0 yr can also be found, but its statistical significance is below the 95% confidence level. The coronal rotation seems to show a weak decreasing trend during the considered time. The dependence of the coronal rotation on solar cycle phase is analyzed. The rotation periods are found to be varying with the solar cycle phase, and they are relatively longer around the minimum year of the solar cycle. The result based on the cross-correlation analysis between the rotation periods and the daily 10.7 cm radio flux indicates that there exists a phase difference of about 5.5 yr between them.

Recently, Chandra and Vats have obtained the yearly period length of the solar coronal rotation cycle by analysing the daily adjusted solar radio flux at the 10.7‐cm wavelength for the years 1947–2009. In this paper, we use the time series (series I) of the yearly period length to investigate the long‐term variation of the rotation of radio emission corona, and we find a weak decreasing trend in the time series. We use the empirical mode decomposition to decompose both the yearly mean value (series II) of the solar radio flux at the 10.7‐cm wavelength and series I into different periodical components. There is a secular trend for each of the two series, and we find a negative correlation in the two trends. The decomposed 11‐yr‐cycle components of the two series show different and complicated periods and there is a phase relation between them. We investigate the cycle‐related variation of the coronal rotation length, and we find that there is no Schwable cycle of statistical significance for the long‐term variation of the rotation cycle length.

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Long-term variations of solar differential rotation and sunspot activity are investigated through re-analyzing the data on parameters of the differential rotation law obtained by Makarov, Tlatov, and Callebaut (1997), Javaraiah, Bertello, and Ulrich (2005a, b), and Javaraiah et al. (2009. Our results indicate that the solar surface rotation rate at the Equator (indicated by the A parameter of the standard solar rotation law) shows a secular decrease since cycle 12 onwards, given by about 1 − 1.5 × 10 −3 (deg day −1 year −1 ). The B parameter of the standard differential rotation law seems to also show a secular decrease since cycle 12 onwards, but of weak statistical significance. The rotation rate averaged on latitudes (0 o − − 40 o ) does not show a secular trend of statistical significance. Moreover, the average sunspot area shows a secular increase of statistical significance since cycle 12 onwards, while a negative correlation is found between the level of sunspot activity (indicated by the average sunspot area) and the solar equatorial rotation in the long run.

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Solar-cycle related variation of differential rotation is investigated through analyzing the rotation rates of magnetic fields, distributed along latitudes and varying with time at the time interval of August 1976 to April 2008. More pronounced differentiation of rotation rates is found to appear at the ascending part of a Schwabe cycle than at the descending part on an average. The coefficient B in the standard form of differential rotation, which represents the latitudinal gradient of rotation, may be divided into three parts within a Schwabe cycle. Part one spans from the start to the 4 th year of a Schwabe cycle, within which the absolute B is approximately a constant or slightly fluctuates. Part two spans from the 4 th to the 7 th year, within which the absolute B decreases. Part three spans from the 7 th year to the end, within which the absolute B increases. Strong magnetic fields repress differentiation of rotation rates, so that rotation rates show less pronounced differentiation, but weak magnetic fields seem to just reflect differentiation of rotation rates. The solar-cycle related variation of solar differential rotation is inferred to the result of both the latitudinal migration of the surface torsional pattern and the repression of strong magnetic activity to differentiation of rotation rates. Sun: rotation-Sun:activity-Sun: surface magnetism 1 *

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The daily sunspot numbers of the whole disk as well as the northern and southern hemispheres from January 1, 1945 to December 31, 2010 are used to investigate the temporal variation of the rotational cycle length through the continuous wavelet transformation analysis method. The auto-correlation function analysis of daily hemispheric sunspot numbers shows that the southern hemisphere rotates faster than the northern hemisphere. The results obtained from the wavelet transformation analysis are: there exists no direct relationship between the variation trend of the rotational cycle length and the variation trend of solar activity in the two hemispheres; the rotational cycle length of both hemispheres has no significant period appearing at the 11 years, but has significant period of about 7.6 years. Analysis concerning the solar cycle dependence of the rotational cycle length shows that in the whole disk and the northern hemisphere acceleration seems to appear before the minimum time of solar activity. Furthermore, the cross-correlation study indicates that the rotational cycle length of the two hemispheres has different phases, and the rotational cycle length of the whole disk as well as the northern and southern hemispheres also has phase shifts with the corresponding solar activity. What's more, the temporal variation of North-South (N-S) asymmetry of the rotational cycle length is also studied; it displays the same variation trend as the N-S asymmetry of solar activity in a solar cycle as well as in the considered time interval, and it has two significant periods of 7.7 and 17.5 years. Moreover, the N-S asymmetry of the rotational cycle length and the N-S asymmetry of solar activity are highly correlated. It's inferred that the northern hemisphere should rotate faster at the beginning of solar cycle 24.