Solar-cycle-related variation of solar differential rotation

MNRAS 433, 521–527 (2013) doi:10.1093/mnras/stt744 Advance Access publication 2013 May 27 Solar-cycle-related variation of solar differential rotation K. J. Li,1,2‹ X. J. Shi,1,3 J. L. Xie,1,3 P. X. Gao,1 H. F. Liang,4 L. S. Zhan5 and W. Feng6 1 National Astronomical Observatories/Yunnan Observatory, CAS, Kunming 650011, China Laboratory of Solar Activity, National Astronomical Observatories, CAS, Beijing 100012, China 3 Graduate School of CAS, Beijing 100863, China 4 Department of Physics, Yunnan Normal University, Kunming 650093, China 5 Jingdezhen Ceramic Institute, Jingdezhen 333001, Jiangxi, China 6 Research Center of Analysis and Measurement, Kunming University of Science and Technology, Kunming 650093, China 2 Key ABSTRACT Solar-cycle-related variation of differential rotation is investigated through analysing the rotation rates of magnetic fields, distributed along latitudes and varying with time at the time interval of 1976 August to 2008 April. 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 1 spans from the start to the fourth year of a Schwabe cycle, within which the absolute B is approximately a constant or slightly fluctuates. Part 2 spans from the fourth to the seventh year, within which the absolute B decreases. Part 3 spans from the seventh 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 be the result of both the latitudinal migration of the surface torsional pattern and the repression of the strong magnetic activity to differentiation of rotation rates. The north–south asymmetry in solar rotation is investigated as well, and the Northern hemisphere should rotate faster than the southern in cycles 21–23. Key words: Sun: activity – Sun: rotation – Sun: surface magnetism. 1 I N T RO D U C T I O N The Sun’s atmosphere displays differential rotation on its disc: the equatorial region of the Sun rotates faster than higher latitude regions (26 d at the solar equator and 30 d at 60◦ latitude) (Howard, Gilman & Gilman 1984; Sheeley, Wang & Nash 1992; Rybak 1994; Altrock 2003; Song & Wang 2005; Le Mouel, Shnirman & Blanter 2007; Fang 2011). To measure angular rotation velocity of the solar atmosphere, three methods have been mainly used: the tracer method, the spectroscopic method and the flux-modulation method (Balthasar & Wohl 1980; Gilman & Howard 1984; Brajsa et al. 2000, 2002; Wohl & Schmidt 2000; Vats et al. 2001; Javaraiah, Bertello & Ulrich 2005; Javaraiah et al. 2009; Wohl et al. 2010; Chandra & Vats 2011b; Vats 2012). Helioseismology measurement can determine solar rotation rate in the solar interior (Howe et al. 2000a,b; Antia & Basu 2001), and the latitudinal shift of solar rotation rate in solar interior determined by helioseismology measurement (Howe et al. 2009) is similar to the torsional oscillation pattern  E-mail: measured by the spectroscopic method (Howard & LaBonte 1980; LaBonte & Howard 1982; Schroter 1985). Up to now, observations and studies on solar differential rotation have taken a great achievement (Howard 1984; Schroter 1985; Snodgrass 1992; Paterno 2010; Vats & Chandra 2011; Vats 2012). However, there are still many aspects, for example the solar-cycle-related and long-term variations of solar rotation rate, unknown at the present (Komm, Howard & Harvey 1993; Ulrich & Bertello 1996; Stix 2002; Li et al. 2011a,b). In this study, we will investigate solar-cycle-related variation of solar rotation rate, using data of the rotation rates of magnetic fields, distributed along latitudes and varying with time at the time interval of 1976 August to 2008 April, and a new explanation is proposed on such a solar-cycle-related variation of the solar rotation rate. 2 S O L A R - C Y C L E - R E L AT E D VA R I AT I O N O F S O L A R D I F F E R E N T I A L ROTAT I O N 2.1 Data Synoptic magnetic maps indicate the distribution of the magnetic fields on the full solar surface, and the rotation rates can be  C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society Accepted 2013 April 27. Received 2013 April 24; in original form 2011 July 21 522 K. J. Li et al. −30 143/10 −20 −10 141/10 0 14 139/10 10 69/5 20 137/10 30 1980 1985 1990 1995 2000 2005 Calendar Year Figure 1. The rotation angular velocities of magnetic fields distributed along latitudes and varying with time, which are obtained by Chu et al. (2010). Latitudes of the Southern hemisphere are negative. The unit shown on the colour bar is degrees per day. derived. Chu et al. (2010) employed Carrington-coordinate synoptic magnetic maps, which are produced by the NSO/Kitt Peak during 1976–2003 and by SOHO/MDI during 2003–2008. They built up a time-longitude stackplot (McIntosh, Willock & Thompson 1991; Japaridze, Gigolashvili & Kukhianidze 2007) at each of latitudes −35◦ to 35◦ . On each stackplot there are many tilted magnetic structures, which clearly reflect the rotation rates, and then they utilized a cross-correlation method to explore the rotation rates from the tilted structures (Chu et al. 2010). Here, Fig. 1 shows the obtained rotation rates distributed along latitudes and varying with time at the time interval of 1976 August to 2008 April, namely from Carrington Rotation (CR) 1645 to 2069. In the figure, latitudes are negative in the Southern hemisphere, and CRs are translated into calendar years. Velocities decrease from the solar equator to high latitudes, and they vary with time, more clearly at relative high latitudes and at the descending part of a sunspot cycle. Fig. 2 shows four isopleth lines of rotation rates, and the corresponding rotation rates are 14.◦ 29, 14.◦ 25, 13.◦ 10 and 13.◦ 85 d−1 in turn from low to high latitudes. Any velocity in the figure is the mean of those at the corresponding latitudes of the two hemispheres. Shown also in the figure are the minimum and maximum times of sunspot cycles. As the figure shows, rotation rates on the average seem higher when the magnetic field is weak, indicating that strong magnetic fields should repress solar differential rotation. 2.2 Solar-cycle-related variation of solar differential rotation The solar differential rotation is usually expressed by the standard formula (Newton & Nunn 1951) ω(φ) = A + Bsin2 φ, Figure 2. Isopleth lines (bold solid lines) of rotation rates. The corresponding rotation rates are 14.◦ 29, 14.◦ 25, 13.◦ 10 and 13.◦ 85 d−1 in turn from low to high la (...truncated)