Large-Scale Magnetic Field of the Sun and Evolution of Sunspot Activity

The evolution of the large-scale magnetic field of the Sun has been studied using an algorithm of tomographic inversion. By analyzing line-of-sight magnetograms, we mapped the radial and toroidal components of the Sun??s large-scale magnetic field. The evolution of the radial and toroidal magnetic field components in the 11-year solar cycle has been studied in a time?Clatitude aspect. It is shown that the toroidal magnetic field of the Sun is causally related to sunspot activity; i.e., the sunspot formation zones drift in latitude and follow the toroidal magnetic fields. The results of our analysis support the idea that the high-latitude toroidal magnetic fields can serve as precursors of sunspot activity. The toroidal fields in the current cycle are anomalously weak and also show a barely noticeable equatorward drift. This behavior of the toroidal magnetic field suggests low activity levels in the current cycle and in the foreseeable future.     

Large-Scale Structure of the Sunspot Activity and the Background Magnetic Field Formation

Large-scale distribution of the sunspot activity of the Sun has been analyzed by using a technique worked out previously (Erofeev, 1997) to study long-lived, non-axisymmetric magnetic structures with different periods of rotation. Results of the analysis have been compared with those obtained by analyzing both the solar large-scale magnetic field and large-scale magnetic field simulated by means of the well-known flux transport equation using the sunspot groups as a sole source of new magnetic flux in the photosphere. A 21-year period (1964–1985) has been examined.The rotation spectra calculated for the total time interval of two 11-year cycles indicate that sunspot activity consists of a series of discrete components (modes) with different periods of rotation. The largest-scale component of the sunspot activity reveals modes with 27-day and 28-day periods of rotation situated, correspondingly, in the northern and southern hemispheres of the Sun, and two modes with rotation periods of about 29.7 days situated in both hemispheres. Such a modal structure of the sunspot activity agrees well with that of the large-scale solar magnetic field. Moreover, the magnetic field distribution simulated with the flux transport equation also reveals the same modal structure. However, such an agreement between the large-scale solar magnetic field and both the sunspot activity and simulated magnetic field is unstable in time; so, it is absent in the northern hemisphere of the Sun during solar cycle No. 20. Thus the sources of magnetic flux responsible for formation of the large-scale, rigidly rotating magnetic patterns appear to be closely connected, but are not identical with the discrete modes of the sunspot activity.     

Analysis of Sunspot Activity Cycles

A nonlinear analysis of the daily sunspot number for each of cycles 10 to 23 is used to indicate whether the convective turbulence is stochastic or chaotic. There is a short review of recent papers considering sunspot statistics and solar activity cycles. The differences in the three possible regimes – deterministic laminar flow, chaotic flow, and stochastic flow – are discussed. The length of data sets necessary to analyze the regimes is investigated. Chaos is described and a chronology of recent results that utilize chaos and fractals to analyze sunspot numbers follows. The parameters necessary to describe chaos – time lag, phase space, embedding dimension, local dimension, correlation dimension, and the Lyapunov exponents – are determined for the attractor for each cycle. Assuming the laminar regime is unlikely if chaos is not indicated in a cycle by the calculations, the regime must be stochastic. The sunspot numbers in each of cycles 10 to 19 indicate stochastic behavior. There is a transition from stochastic to chaotic behavior of the sunspot numbers in cycles 20, 21, 22, and 23. These changes in cycles 20 – 23 may indicate a change in the scale of turbulence in the convection zone that could result in a change in the convective heat transfer and a change in the size of the convection region for these four cycles.     

The Shape of Sunspot Cycles Described by Monthly Sunspot Areas

In this paper, we used the same four-parameter function as Hathaway, Wilson, and Reichmann (1994) proposed and studied the temporal behavior of sunspot cycles 12–22. We used the monthly averages of sunspot areas and their 13-point smoothed data. Our results show the following. (1) The four-parameter function may reduce to a function of only two parameters. (2) As a cycle progresses, the two-parameter function can be accurately determined after 4–4.5 years from the start of the cycle. A good prediction can be made for the timing and size of the sunspot maximum and for the behavior of the remaining 5–10 years of the cycle. (3) The solar activity in the remaining and forthcoming years of cycle 23 is predicted. (4) The smoothed monthly sunspot areas are more suitable to be employed for prediction at the maximum and the descending period of a cycle, whereas at the early period of a cycle the (un-smoothed) monthly data are more suitable.     

The Relationships of Sunspot Magnetic Field Strength with Sunspot Area, Umbral Area and Penumbra-Umbra Radius Ratio

Stokes profile inversion is very important to get the information on the vector magnetic field. Because the magnetic fields cannot be directly observed, adopting Stokes spectrum analysis to obtain vector magnetic field has become the major technique recently. Therefore, by Stokes profile inversion, we obtained vector magnetic fields of two layers based on the numerical solution (DELO solution, ReEs et al., 1989) to the polarized radiative transfer equation. We analyze the relationships of sunspot magnetic field strength with sunspot area, umbral area and penumbra-umbra radius ratio. By statistical research, it is found that the field strengths of the upper layer and the lower one decrease with the increasing penumbra-umbra radius ratio, and that the logarithmic expression is able to fit well the relationship between the maximum field strength of the upper layer and the sunspot area. Furthermore, we verify the result obtained by Ringnes and Jensen (1961) about the relationship between the maximum magnetic field strength and the umbral area, and the result obtained by Antalová (1991) of the relationship between the field strength and the penumbra-umbra radius ratio.     

A Stochastic Prediction Model for the Sunspot Cycles

A stochastic prediction model for the sunspot cycle is proposed. The prediction model is based on a modified binary mixture of Laplace distribution functions and a moving-average model over the estimated model parameters. A six-parameter modified binary mixture of Laplace distribution functions is used for the modeling of the shape of a generic sunspot cycle. The model parameters are estimated for 23 sunspot cycles independently, and the primary prediction-model parameters are derived from these estimated model parameters using a moving-average stochastic model. A correction factor (hump factor) is introduced to make an initial prediction. The hump factor is computed for a given sunspot cycle as the ratio of the model estimated after the completion of a sunspot cycle (post-facto model) and the prediction of the moving-average model. The hump factors can be applied one at a time over the moving-average prediction model to get a final prediction of a sunspot cycle. The present model is used to predict the characteristics of Sunspot Cycle 24. The methodology is validated using the previous Sunspot Cycles 21, 22, and 23, which shows the adequacy and the applicability of the prediction model. The statistics of the variations of sunspot numbers at high solar activity are used to provide the lower and upper bound for the predictions using the present model.     

The Sun’s Large-Scale Magnetic Field and Its Long-Term Evolution

The Sun’s large-scale external field is formed through the emergence of magnetic flux in active regions and its subsequent dispersal over the solar surface by differential rotation, supergranular convection, and meridional flow. The observed evolution of the polar fields and open flux (or interplanetary field) during recent solar cycles can be reproduced by assuming a supergranular diffusion rate of 500 – 600 km² s⁻¹ and a poleward flow speed of 10 –20 m s⁻¹. The nonaxisymmetric component of the large-scale field decays on the flow timescale of ∼1 yr and must be continually regenerated by new sunspot activity. Stochastic fluctuations in the longitudinal distribution of active regions can produce large peaks in the Sun’s equatorial dipole moment and in the interplanetary field strength during the declining phase of the cycle; by the same token, they can lead to sudden weakenings of the large-scale field near sunspot maximum (Gnevyshev gaps). Flux transport simulations over many solar cycles suggest that the meridional flow speed is correlated with cycle amplitude, with the flow being slower during less active cycles.     

The Relationship Between Solar Activity and the Large-Scale Axisymmetric Magnetic Field

To investigate the relationship between solar activity and the large-scale axisymmetric magnetic field of the Sun, we inferred from sunspot data over the period 1964–1985 a latitude–time distribution of magnetic field associated with active regions. This has been done allowing for both bipolar structure of the active regions and inclination of their axes to parallels of latitude, so the inferred magnetic field characterizes latitudinal separation of magnetic polarities which might be related to the large-scale magnetic field of the Sun according to the Babcock–Leighton model. The inferred magnetic field, A _z, is compared with the longitude-averaged (zonal) magnetic field of the Sun, B _z, derived from series of magnetograms obtained at Mount Wilson Observatory in the years 1964–1976, and at Kitt Peak National Observatory during the period from 1976 to 1985. The inferred magnetic field, A _z, exhibits a complex structure distribution of magnetic polarities with respect to latitude and time. Apart from concentration of the different polarity magnetic fields inside the high- and low-latitude portions of the sunspot belts, bipolar active regions produce an intensive, shorter-scale component of the magnetic field which varies on the time scale of about 2 years. Such a short-term variation of A _z reveals substantial correlation with the short-term component of B _z which has the form of the poleward-drifting streams of magnetic field. Most significant correlation takes place between the short-term variations of A _z occurring at latitudes below 20° and those of the large-scale magnetic fields occurring at middle latitudes of 40–50°. Moreover we analyze harmonic coefficients a _l and b _l obtained by expanding A _z and B _z into series in terms of the spherical harmonics. Power spectra of the time-dependent harmonic coefficients indicate that both A _z and B _z reveal a number of resonant modes which oscillate either with the 22-year period in the case of the anti-symmetric (odd-l) modes or with periods of about 2 years in the case of the symmetric (even-l) modes, but the resonant modes of A _z have significantly larger values of the spherical harmonic degree l (and, hence, smaller spatial scales) as compared to those of B _z. It is found that there is a close relationship between the harmonic coefficients b _l and a _m for which either m–l16 (even l=4,...,10) or m–l=4 (odd l=5,...,15).     

无力场模型黑子上空的纵向磁场梯度

本文按常α无力场模型计算了1980年10月23日Boulder 2744活动区前导黑子的纵向磁场随高度的变化,并与用CIV 1548谱线观测得到的色球一日冕过渡区的磁场资料相结合,求得CIV 1548发射区的有效高度。这些结果与文献[4]中对同一黑子用势场模型推求的结果有很大差别。从而表明,势场和无力场在某些方面导致的结果是极不相同的。鉴于观测已表明活动区上空存在电流的事实,在活动区磁场的模拟中,特别是在强扭曲活动区磁场的计算中,应当避免采用势场,而尽可能采用无力场模型。     

A Bayesian Analysis of the Correlations Among Sunspot Cycles

Sunspot numbers form a comprehensive, long-duration proxy of solar activity and have been used numerous times to empirically investigate the properties of the solar cycle. A number of correlations have been discovered over the 24 cycles for which observational records are available. Here we carry out a sophisticated statistical analysis of the sunspot record that reaffirms these correlations, and sets up an empirical predictive framework for future cycles. An advantage of our approach is that it allows for rigorous assessment of both the statistical significance of various cycle features and the uncertainty associated with predictions. We summarize the data into three sequential relations that estimate the amplitude, duration, and time of rise to maximum for any cycle, given the values from the previous cycle. We find that there is no indication of a persistence in predictive power beyond one cycle, and we conclude that the dynamo does not retain memory beyond one cycle. Based on sunspot records up to October 2011, we obtain, for Cycle 24, an estimated maximum smoothed monthly sunspot number of 97±15, to occur in January??C?February 2014 ± six months.     

The Global Solar Magnetic Field Through a Full Sunspot Cycle: Observations and Model Results

Based on 11 years of SOHO/MDI observations from the cycle minimum in 1997 to the next minimum around 2008, we compare observed and modeled axial dipole moments to better understand the large-scale transport properties of magnetic flux in the solar photosphere. The absolute value of the axial dipole moment in 2008 is less than half that in the corresponding cycle-minimum phase in early 1997, both as measured from synoptic maps and as computed from an assimilation model based only on magnetogram data equatorward of 60° in latitude. This is incompatible with the statistical fluctuations expected from flux-dispersal modeling developed in earlier work at the level of 7 – 10 σ. We show how this decreased axial dipole moment can result from an increased strength of the diverging meridional flow near the Equator, which more effectively separates the two hemispheres for dispersing magnetic flux. Based on the combination of this work with earlier long-term simulations of the solar surface field, we conclude that the flux-transport properties across the solar surface have changed from preceding cycles to the most recent one. A plausible candidate for such a change is an increase of the gradient of the meridional-flow pattern near the Equator so that the two hemispheres are more effectively separated. The required profile as a function of latitude is consistent with helioseismic and cross-correlation measurements made over the past decade.     

Duration of Polar Activity Cycles and Their Relation to Sunspot Activity

We have defined the duration of polar magnetic activity as the time interval between two successive polar reversals. The epochs of the polarity reversals of the magnetic field at the poles of the Sun have been determined (1) by the time of the final disappearance of the polar crown filaments and (2) by the time between the two neighbouring reversals of the magnetic dipole configuration (l=1) from the H synoptic charts covering the period 1870–2001. It is shown that the reversals for the magnetic dipole configuration (l=1) occur on an average 3.3±0.5 years after the sunspot minimum according to the H synoptic charts (Table I) and the Stanford magnetograms (Table III). If we set the time of the final disappearance of the polar crown filaments (determined from the latitude migration of filaments) as the criterion for deciding the epoch of the polarity reversal of the polar fields, then the reversal occurs on an average 5.8±0.6 years from sunspot minimum (last column of Table I). We consider this as the most reliable diagnostic for fixing the epoch of reversals, as the final disappearance of the polar crown filaments can be observed without ambiguity. We show that shorter the duration of the polar activity cycle (i.e., the shorter the duration between two neighbouring reversals), the more intense is the next sunspot cycle. We also notice that the duration of polar activity is always more in even solar cycles than in odd cycles whereas the maximum Wolf numbers W _\max is always higher for odd solar cycles than for even cycles. Furthermore, we assume there is a secular change in the duration of the polar cycle. It has decreased by 1.2 times during the last 120 years.     

Determination of the Coronal and Interplanetary Magnetic Field Strength and Radial Profiles from Large-Scale Photospheric Magnetic Fields

We propose a new model for the magnetic field at different distances from the Sun during different phases of the solar cycle. The model depends on the observed large-scale non-polar (\({\pm}\, 55^{\circ }\)) photospheric magnetic field and on the magnetic field measured at polar regions from \(55^{\circ }\) N to \(90^{\circ }\) N and from \(55^{\circ }\) S to \(90^{\circ }\) S, which are the visible manifestations of cyclic changes in the toroidal and poloidal components of the global magnetic field of the Sun. The modeled magnetic field is determined as the superposition of the non-polar and polar photospheric magnetic field and considers cycle variations. The agreement between the model predictions and magnetic fields derived from direct in situ measurements at different distances from the Sun, obtained with different methods and at different solar activity phases, is quite satisfactory. From a comparison of the magnetic fields as observed and calculated from the model at 1 AU, we conclude that the model magnetic field variations adequately explain the main features of the interplanetary magnetic field (IMF) radial, \(B_{\mathrm{x}}\), component cycle evolution at Earth’s orbit. The modeled magnetic field averaged over a Carrington rotation (CR) correlates with the IMF \(B_{\mathrm{x}}\) component also averaged over a CR at Earth’s orbit with a coefficient of 0.691, while for seven CR-averaged data, the correlation reaches 0.81. The radial profiles of the modeled magnetic field are compared with those of already existing models. In contrast to existing models, ours provides realistic magnetic-field radial distributions over a wide range of heliospheric distances at different cycle phases, taking into account the cycle variations of the solar toroidal and poloidal magnetic fields. The model is a good approximation of the cycle behavior of the magnetic field in the heliosphere. In addition, the decrease in the non-polar and polar photospheric magnetic fields is shown. Furthermore, the magnetic field during solar cycle maxima and minima decreased from Cycle 21 to Cycle 24. This implies that both the toroidal and poloidal components, and therefore the solar global magnetic field, decreased from Cycle 21 to Cycle 24.     

Tilt of Sunspot Bipoles in Solar Cycles 15 to 24

We use recently digitized sunspot drawings from Mount Wilson Observatory to investigate the latitudinal dependence of tilt angles of active regions and its change with solar cycle. The drawings cover the period from 1917 to present and contain information as regards polarity and strength of magnetic field in sunspots. We identified clusters of sunspots of same polarity, and used these clusters to form “bipole pairs”. The orientation of these bipole pairs was used to measure their tilts. We find that the latitudinal profile of tilts does not monotonically increase with latitude as most previous studies assumed, but instead, it shows a clear maximum at about 25?–?30 degree latitudes. Functional dependence of tilt (\(\gamma\)) on latitude (\(\varphi\)) was found to be \(\gamma= (0.20\pm0.08) \sin(2.80 \varphi) + (-0.00\pm0.06)\). We also find that latitudinal dependence of tilts varies from one solar cycle to another, but larger tilts do not seem to result in stronger solar cycles. Finally, we find the presence of a systematic offset in tilt of active regions (non-zero tilts at the equator), with odd cycles exhibiting negative offset and even cycles showing the positive offset.     

Cyclic and Long-Term Variation of Sunspot Magnetic Fields

Measurements from the Mount Wilson Observatory (MWO) were used to study the long-term variations of sunspot field strengths from 1920 to 1958. Following a modified approach similar to that presented in Pevtsov et al. (Astrophys. J. Lett. 742, L36, 2011), we selected the sunspot with the strongest measured field strength for each observing week and computed monthly averages of these weekly maximum field strengths. The data show the solar cycle variation of the peak field strengths with an amplitude of about 500?–?700 gauss (G), but no statistically significant long-term trends. Next, we used the sunspot observations from the Royal Greenwich Observatory (RGO) to establish a relationship between the sunspot areas and the sunspot field strengths for cycles 15?–?19. This relationship was used to create a proxy of the peak magnetic field strength based on sunspot areas from the RGO and the USAF/NOAA network for the period from 1874 to early 2012. Over this interval, the magnetic field proxy shows a clear solar cycle variation with an amplitude of 500?–?700 G and a weaker long-term trend. From 1874 to around 1920, the mean value of magnetic field proxy increases by about 300?–?350 G, and, following a broad maximum in 1920?–?1960, it decreases by about 300 G. Using the proxy for the magnetic field strength as the reference, we scaled the MWO field measurements to the measurements of the magnetic fields in Pevtsov et al. (2011) to construct a combined data set of maximum sunspot field strengths extending from 1920 to early 2012. This combined data set shows strong solar cycle variations and no significant long-term trend (the linear fit to the data yields a slope of ??0.2±0.8 G?year^?1). On the other hand, the peak sunspot field strengths observed at the minimum of the solar cycle show a gradual decline over the last three minima (corresponding to cycles 21?–?23) with a mean downward trend of ≈?15 G?year^?1.     

The Relationship Between Kinematics and Spatial Structure of the Large-Scale Solar Magnetic Field

The rotation of large-scale solar magnetic fields has been investigated by analysing a 20-yr series of synoptic maps of the radial magnetic field. For this purpose, a specially adapted method of spectral analysis was used. We calculated rotation spectra of the magnetic field as functions of the rotation period, heliographic latitude, and longitudinal wave number, k. These spectra reveal the existence of a number of discrete, rigidly rotating components (modes) of the magnetic field, whose rotation periods lie in the wide range from 26.5 to 30.5 days. The significant spectral maxima lie in the (rotation period–latitude) plane close to the curve that represents the differential rotation of small-scale magnetic features. For the first harmonic of the magnetic field (k=1) the properties of the rotation spectra are consistent with those reported by Antonucci, Hoeksema, and Scherrer (1990). However, the distribution of the rigidly rotating modes over rotation period and their latitudinal structure change systematically with the harmonic number k. As k increases, the mean distance P in rotation period between the modes decreases, from 1.2 days for k=1 to 0.3–0.5 days for k=4. This decreasing period separation is accompanied by a decrease of the characteristic latitude separation between the mode maxima. The latitudinal and longitudinal discrete spatial scales of the non-axisymmetric magnetic field appear to be connected with each other, as well as with the temporal scale P.     

Changing Relationships Between Sunspot Number,Total Sunspot Area and $F_{10.7}$ in Cycles 23 and 24

This article is an update of a study (Tapping and Valdès in Solar Phys. 272, 337, 2011) made in the early part of Cycle 24 using an intercomparison of various solar activity indices (including sunspot number and the 10.7 cm solar radio flux), in which it was concluded that a change in the relationship between photospheric and chromospheric/coronal activity took place just after the maximum of Cycle 23 and continued into Cycle 24. Precursors (short-term variations) were detected in Cycles 21 and 22. Since then the sunspot number index data have been substantially revised. This study is intended to be an update of the earlier study and to assess the impact of the revision of the sunspot number data upon those conclusions. This study compares original and revised sunspot number, total sunspot area, and 10.7 cm solar radio flux. The conclusion is that the transient changes in Cycles 21 and 22, and the more substantial change in Cycle 23, remain evident. Cycle 24 shows indications that the deviation was probably another short-term one.     

太阳黑子的世纪周期及对24、25活动周的预报

太阳活动除了具有公认的11 a周期以外,还存在着一个80～120 a变化的世纪周期,也称为Gleissberg周期.使用傅里叶变换和小波分析的方法,分析了1700～2008年的年均黑子数世纪周期的变化规律.得到结果:在太阳活动世纪周期的低谷期,所对应11 a太阳周的极大年和极小年的黑子数目都比其他太阳周的低.在这300多年里,世纪周期的周期长度也有变化.由世纪周期的变化趋势,预测第24、25太阳活动周将处于世纪周期的低谷期.通过对以前3个世纪周期的谷期黑子数求平均的方法,得到第24,25太阳周极大年年均黑子数为63.6±21.1,极小年的为2.2±2.1.这些结果有助于理解当前太阳活动反常宁静这一现象.     

Sunspot Rotations Derived from Magnetic and Velocity Fields Observations

The rotation of sunspot penumbrae has been investigated on the longitudinal magnetic and velocity fields, observed in the photospheric line Fe i λ5253 Å of five lone sunspots. We reconstructed the entire vectors of both fields from their line-of-sight components. All three components of both vectors revealed that the rotation of the sunspots was, in fact, a torsional oscillation. All components of each sunspot had the same rotational period. The penumbrae oscillation periods were distributed in the range from 3.4 days to 7.7 days. The phase of the velocity azimuthal component oscillation was ahead of the phases of all other components of both vectors. If the penumbra plasma density had been equal to the photospheric plasma density (10^?7 g cm^?3) then the oscillation magnetic energy of the components exceeded their kinetic energy approximately by a factor of 10–200. The obtained results led to the conclusion that these oscillations were constrained.     

磁光效应与太阳黑子磁场结构

本文在考虑磁光效应条件下,根据对斯托克斯参数转移方程组求得的数值解,计算了单极太阳黑子的线偏振讯号的单色像,并与美国马歇尔空间飞行中心的观测资料进行了对比,结果表明,径向黑子磁场模型给出与观测相似的单色像,而旋涡形模型导致与观测有显著差异的图像。因此可以认为径向模型更接近于实际情况。