SOLAR ACTIVE LONGITUDES AND THEIR ROTATION

LIYUN ZHANG Department of Physics University of Oulu Finland

Academic dissertation to be presented, with the permission of the Faculty of Science of the University of Oulu, for public discussion in the Auditorium L6, Linnanmaa, on 30th November, 2012, at 12 o’clock noon.

REPORT SERIES IN PHYSICAL SCIENCES Report No. 78

OULU 2012 UNIVERSITY OF OULU

Supervisors Prof. Kalevi Mursula University of Oulu, Finland Prof. Ilya Usoskin Sodankylä Geophysical Observatory, University of Oulu, Finland Opponent Dr. Rainer Arlt Leibniz-Institut für Astrophysik Potsdam, Germany Reviewers Prof. Roman Brajša Hvar Observatory, Faculty of Geodesy, University of Zagreb, Croatia Assoc. Prof. Dmitri Ivanovitch Ponyavin St. Petersburg State University, St. Petergoff, Russia Custos Prof. Kalevi Mursula University of Oulu, Finland ISBN 978-952-62-0025-5 ISBN 978-952-62-0026-2 (PDF) ISSN 1239-4327 Oulu University Press Oulu 2012

Zhang, Liyun:Solar active longitudes and their rotation

Department of Physical Sciences, University of Oulu, Finland.Report No. 78 (2012)

Abstract

In this thesis solar active longitudes of X-ray flares and sunspots are studied. The factthat solar activity does not occur uniformly at all heliographic longitudes was noticedby Carrington as early as in 1843. The longitude ranges where solar activity occurspreferentially are called active longitudes. Active longitudes have been found in variousmanifestations of solar activity, such as sunspots, flares, radio emission bursts, surfaceand heliospheric magnetic fields, and coronal emissions. However, the active longitudesfound when using different rigidly rotating reference frames differ significantly from eachother. One reason is that the whole Sun does not rotate rigidly but differentially atdifferent layers and different latitudes. The other reason is that the rotation of the Sunalso varies with time.

Earlier studies used a dynamic rotation frame for the differential rotation of the Sunand found two persistent active longitudes of sunspots in 1878-1996. However, the migra-tion of active longitudes with respect to the Carrington rotation was treated there rathercoarsely. We improved the accuracy of migration to less than one hour. Accordingly,not only the rotation parameters for each class of solar flares and sunspots are foundto agree well with each other, but also the non-axisymmetry of flares and sunspots issystematically increased.

We also studied the long-term variation of solar surface rotation. Using the improvedanalysis, the spatial distribution of sunspots in 1876-2008 is analyzed. The statisticalevidence for different rotation in the northern and southern hemispheres is greatly im-proved by the revised treatment. Moreover, we have given consistent evidence for theperiodicity of about one century in the north-south difference.

Keywords: Solar activity, sunspots, X-ray flares, solar rotation, active longitudes.

This thesis is dedicated to my biological parents and myadoptive parents for all their love and support! They havebeen and will always be great sources of encouragement

and inspiration throughout my life!

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Acknowledgements

This work has been carried out in the Department of Physics at the Universityof Oulu, Finland. First, I would like to express my deepest gratitude to Prof.Ilya Usoskin. He very enthusiastically answered my questions at our first meetduring the Beijing COSPAR 2006. He has been supporting me with my studyand work ever since. He also guided me to write my first scientific paper in 2006.He encouraged me to come to Oulu to get valuable work experience under theguidance of Prof. Kalevi Mursula and himself. In the beginning of my first visitto Oulu, Ilya offered me plenty of help and guidance for managing the new dailylife, including how to wait for a bus and how to use the knife and fork at lunch,too much to mention.

Next, I wish to express my greatest gratitude to my principal supervisor, Prof.Kalevi Mursula who always has open mind and new ideas. He has always beengiving me plenty of support and encouragement to deal with the problems I havehad during doing the thesis. Accordingly, the scientific significance of the thesishas been greatly improved. I also sincerely appreciate his fatherly advices on mylife.

I would like to express my deepest gratitude to my Beijing chief Prof. HuaningWang, a great scientist in solar physics and solar activity prediction. He has beenguiding me through my science career since the very beginning and giving mecontinuous support. He always reminds me to find out what is the physics behindthe phenomenon that I see. The other members of my Beijing group are alsodeeply acknowledged, especially Prof. Zhanle Du, Xin Huang, Jian Gao, YanmeiCui and Rong Li, who greatly share my study, work and life.

I am very grateful for the funding from the Department of Physics of the Uni-versity of Oulu, the Academy of Finland, the European Commission’s SeventhFramework Programme for eHeroes project, and the Key Laboratory of Solar Ac-tivity, National Astronomical Observatories, Chinese Academy of Sciences.

I fully acknowledge the pre-examiners of my thesis Prof. Roman Brajsa andAssoc. Prof. Dimitri Ponyavin for their valuable comments and advice. I would

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also like to express my sincere appreciation to Dr. Rainer Arlt who agreed to sparehis time to be the opponent.

I wish to express my warmest gratitude to all the other members in the SpacePhysics group for their friendship and help, Reijo Rasinkangas, Leena Kalliopuska,Olesya Yakovchouk, Alessandra Pacini, three Virtanen brothers, Timo Asikainen,Petri Kalliomaki, Tomi Karppinen, Lauri Holappa, Ville Maliniemi, Anita Aikio,Ritva Kuula, Tuomo Nygren, Kari Kalla, Timo Pitkanen, Lei Cai, Perttu Kantolaand Raisa Leussu. Members of the Geophysics group are also acknowledged fortheir warm companionship during the coffee break.

In the end I would like to thank my dear sister and brother for their constantsupport and love during my study and my life. I definitely must thank my belovedhusband Ari Ronkainen for everything. He has been strongly supporting me formy work and for my hobbies. I would also like to thank my husband’s parents andrelatives for their love and support.

Oulu, November 30, 2012

Liyun Zhang

Original publications

This thesis consists of an introduction and four original papers.

I L. Zhang, K. Mursula, I. G. Usoskin, and H. N. Wang, Global analysis ofactive longitudes of solar X-ray flares, J. Atmos. Sol. Terr. Phys., 73, 258-263, 2011.

II L. Zhang, K. Mursula, I. G. Usoskin, and H. N. Wang, Global analysis ofactive longitudes of sunspots, Astron. Astrophys., 529, A23, 2011.

III L. Zhang, K. Mursula, I. Usoskin, H. Wang, and Z. Du, Long-term variationof solar surface differential rotation, Bulletin of Astronomical Society of India,ASI Conference Series, 2, 175, 2011.

IV L. Zhang, K. Mursula, and I. Usoskin, Consistent long-term variation in thehemispheric asymmetry of solar rotation, Astron. Astrophys., submitted.

In the text, original papers are referred to using roman numerals I-IV.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Original publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112. Solar activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1. Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.1. Wolf sunspot numbers . . . . . . . . . . . . . . . . . . . . . 172.1.2. Group sunspot numbers . . . . . . . . . . . . . . . . . . . . 192.1.3. Sunspot areas and positions . . . . . . . . . . . . . . . . . . 202.1.4. Cyclicities in solar activity . . . . . . . . . . . . . . . . . . 212.1.5. Grand minima and maxima . . . . . . . . . . . . . . . . . . 25

2.2. Solar flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1. Classification of solar flares . . . . . . . . . . . . . . . . . . 292.2.2. Phases of a typical flare . . . . . . . . . . . . . . . . . . . . 312.2.3. Radio emissions during solar flares . . . . . . . . . . . . . . 342.2.4. Flare models . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3. Coronal mass ejections . . . . . . . . . . . . . . . . . . . . . . . . . 392.4. Solar energetic particles acceleration . . . . . . . . . . . . . . . . . 41

3. Solar rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1. Heliographic coordinates . . . . . . . . . . . . . . . . . . . . . . . . 433.2. Surface rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3. Internal rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4. Torsional oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5. Meridional circulation . . . . . . . . . . . . . . . . . . . . . . . . . 51

4. Active latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555. Active longitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1. History of active longitude research . . . . . . . . . . . . . . . . . . 585.2. Flip-flop phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . 695.3. Dynamo models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4. North-South asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4.1. N-S asymmetry in solar activity . . . . . . . . . . . . . . . 75

5.4.2. N-S asymmetry in solar rotation . . . . . . . . . . . . . . . 776. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1. Introduction

As is well known, the Sun is the source of energy that supports the existence of lifeon the Earth through radiation, which was originally recognized as visible sunlight,while solar electromagnetic radiation covers infrared, visible, and ultraviolet light.Most of the time the Sun appears as a quiet star to the naked-eye, but it isfar more active when observed through telescopes, especially through the modernspace-borne instruments with high temporal and spatial resolution. Even the quietSun appears very active in the observations of the newest space solar telescope -Solar Dynamics Observatory (SDO). Small scale tornadoes or cyclones are foundto occur all over the quiet Sun with rotating magnetic fields [Zhang and Liu,2011] (see the left panel of Fig. 1.1). Later phases of cyclones are observed tobe associated with microflares. A huge tornado can be observed before a coronalmass ejection (CME) (see the right panel of Fig. 1.1).

There are gradually changing activities and bursting activities on the Sun. Theformer refer to activities which vary gradually in size and energy and last a rel-atively long time. These include sunspots, plages, faculae, quiet filaments andprominences, coronal holes and streamers, solar wind, and the general radiation.The latter refer to explosive events with sudden and huge energy release. Theseevents include, e.g., flares, CMEs, and explosive filaments and prominences.

Solar activity affects the near-Earth space in many different ways. Variation insolar conditions at short time scales, e.g., from a few seconds to days is involvedwith space weather, while the long-term conditions, from several months to onesolar cycle and longer, relate to space climate. Space weather effects are mainlycaused by explosive transient solar events, i.e., flares and CMEs. The varioustypes of radiation released in a flare can be absorbed in the Earth’s atmosphere.The heated atmosphere might drag a spacecraft to a lower orbit in altitude. Solarenergetic particles (SEPs), which are accelerated up to relativistic velocities in aflare or by CME shock waves, can cause a disruption of technical systems andmay threaten the life and health of astronauts working in the space outside theprotection of the Earth’s magnetosphere. Gigantic clouds of ionized gas ejectedin a CME can cause a series of geomagnetic effects when they reach and hit theEarth’s magnetosphere. These effects can cause malfunctions in satellite systems.They cause the ionosphere to more effectively absorb high frequency radio waves,

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Y (

arc

sec)

-280

-580

-600

-260X (arcsec)

Fig. 1.1. Tornadoes on the Sun observed with SDO/AIA 171A EUV. Left panel: an

example of small-scale tornadoes observed in the quiet Sun on 20 July 2010 [Zhang and

Liu, 2011]. Right panel: A huge tornado observed before a CME with the width as

large as five Earths on 25 September 2011. [Credit: NASA/SDO/AIA/Aberystwyth

University/Li/Morgan/Leonard; http://users.aber.ac.uk/xxl/tornado.htm.]

thus affecting radio transmission or even making it impossible. The first solarinduced technological damage was noticed in the telegraph transmission due to awhite flare observed by Carrington and Hodgson in 1859. This event was widelyspread through newspapers and articles, which caught much public attention andbecame the first famous space weather event.

With more and more utilization of space and technical systems in our daily life,the economic and societal losses due to powerful solar explosive events have becomemore and more serious. In order to lessen this damage, sciences and techniquesinvolved in these processes have to be improved. On one hand, we have to seek outsignatures of indication of a powerful solar event and the connections of solar eventsand the space and near-Earth environment so that we can identify the geoeffectiveevents from those which are not geoeffective. This study has developed into spaceweather and forecasting sciences. Although the Solar-terrestrial physics startedlong ago, a large amount of connections and physical mechanisms involved withsolar activity and space effects are still open. The signatures of a geoeffective eventhave not been surely identified yet. On the other hand, we need to estimate howserious the effects from a solar event can be so that the corresponding protectivemeasures for technical systems and humans can be found.

Through observations of space-borne instruments, especially in the last decade,the appearance and evolution of solar activity and its effect on the solar-terrestrialenvironment has been roughly established. Space weather, nowadays, deals withactivity conditions on the Sun and their effects on the near-Earth space, especially

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what might be hazardous to the technological systems and endanger humans’ lifeand health both in the spacecraft and on the ground.

At the present time the parameters of solar wind (the material in the interplan-etary space) can only be observed at the first Lagrangian point (L1 ∼1.5·106 kmaway to the Earth, 1/100 of the Sun-Earth distance). This leaves us only 30 to60 minutes before the plasma reaches the Earth. ACE (Advanced CompositionExplorer launched in 1997), SOHO (Solar and Heliospheric Observatory launchedin 1995) and WIND (Global Geospace Science WIND moved to L1 in 2004) at theL1 point measure real time solar wind parameters, such as the magnetic field, thevelocity density and temperature of electrons and protons.

International Space Environment Service (ISES) consisting of 13 Regional War-ing Centers (RWCs) all over the world, and the European Space Agency (ESA)collect space-borne and ground-based space-weather related observations and fore-casts and provide the data to users. The Space Weather Prediction Center ofNOAA (National Oceanic and Atmospheric Administration of USA) performs 3-day space weather forecasts and plots of the observational data (http://www.swpc.noaa.gov/today.html).

The magnetic field in the interplanetary space has a spiral structure. This leadsthat only those charged particles accelerated in the flares occurring close to thewestern limb of the solar disc may hit the Earth. Magnetic clouds from CMEs withinclosed magnetic structure propagate radially from the Sun to the interplanetaryspace. Thus, CMEs may reach the near-Earth space when they appear close tothe center of solar disc. Those events which occur on the back side of the Sun arenot geo-effective. Therefore, the location of eruptions on the Sun is an importantfactor to space weather.

It was noted long ago that solar activity does not occur uniformly at all longi-tudes on the Sun. Those longitude ranges where solar activity appear more oftenthan elsewhere are called active longitudes. Active longitudes (ALs) were foundin various manifestations of solar activity, such as sunspots, flares, surface mag-netic fields. However, the suggested ALs differ with each other significantly inrigidly rotating reference frame. This is because the Sun does not rotate rigidlybut differentially with latitude and depth, showing the so-called surface and radialdifferential rotation. Accordingly, different types of solar activity appearing atdifferent height in the solar atmosphere, e.g., sunspots at solar surface and CMEsin the high corona, rotate at different rates. And the rotation of one type of solaractivity may change due to its latitudinal variation in a solar cycle.

Taking solar surface differential rotation into account Usoskin et al. [2005] in-troduced a dynamic differentially rotating reference frame. In this frame, theshift of ALs due to the surface differential rotation was removed. Two persis-tent active longitudes of sunspots were found to last for 11 cycles. Using similarreference system two ALs of X-ray flares were found to exist for 3 cycles. Thenon-axisymmetry of solar flares is much higher than that of sunspots. Especially,the non-axisymmetry of X-class flares can reach 0.55, suggesting that 77.5% ofpowerful flares occur in the two active longitudes. This implies the importance ofprediction of active longitudes of solar flares. However, the rotation parametersobtained for each class of solar flares differ with each other significantly.

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In order to find out the source of the discrepancy among the rotation parametersfor sunspots and each class of flares, the locations of sunspots and X-ray flares inlast three solar cycles were reanalyzed in this thesis.

Rotation rate of the Sun varies also with time. Increasing evidence [Howard andHarvey, 1970; Knaack et al., 2005; Balthasar, 2007] suggests that the rotation rateof solar surface changes from cycle to cycle, even within one cycle. The rotationrate in the northern hemisphere was found to vary differently with the southernhemisphere showing N-S asymmetry. The suggested faster rotating hemispheresin each solar cycle by different authors disagree with each other. The locations ofsunspots in 1876-2008 are analyzed in this thesis in order to study the long-termvariation in solar surface rotation.

2. Solar activity

2.1. Sunspots

Sunspots appear visibly as dark spots compared to surrounding regions. Therefore,they can easily be observed even by naked-eye [Keller and Friedli, 1992], and areknown as the one of several active solar phenomena which was observed earliest.For instance, several dark spots in one group (AR10930) can be seen in the solardisc observed on Dec 11, 2006 by SOHO (see Fig. 2.1 ). Although the details ofsunspot generation are still a matter of research, sunspots are generally thought tobe generated by toroidal magnetic fields lying below the photosphere and broughtto the surface by buoyancy [Kivelson and Russell, 1995; Rigozo et al., 2001].

Registered sunspot observations have been dated at least back to the fourthcentury BC. Irregular naked-eye observations of sunspot activity can be found inChinese dynastic chronicles since 28 BC [Ding, 1978; Polygiannakis et al., 1996;Mundt et al., 1991; Li et al., 2002] or even 165 BC [Wittmann and Xu, 1987].These are Chinese oracle bones dating from before 1000 BC which record sunspots[Hsu, 1972; Price et al., 1992]. It was shown that the oriental historic data reallyreflect some features of sunspot activity such as 11-year and century scale cycles,deep Maunder-type minima of solar activity [Ogurtsov et al., 2002]. However,these early records have some natural shortcomings: First, it is very likely thatancient astronomers often mixed real sunspots with other celestial or meteorologi-cal phenomena; Second, ancient sunspot observations were not systematic and, asa result, are non-uniform in time; Third, some climatic effects may be present indifferent ancient sunspot series [Willis et al., 1980].

Sunspots were first telescopically observed in 1610 by the English astronomerThomas Harriot and the German (Frisian) father and son, David and JohannesFabricius. David and Johannes Fabricius noted the rotation of sunspots and theSun and published a description on the rotation in 1611. The rotation of the Sunand sunspots was independently discovered by Christoph Scheiner and GalileoGalilei as well in 1610 by observing sunspots continuously for a few days. Theyboth published their observations in 1612. Using an improved telescope, ChristophScheiner measured the rotation rate of sunspots near the equator in 1630. Unfor-tunately, not many sunspots were recorded in the following hundred years since

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Fig. 2.1. Sunspots observed with SOHO/MDI continuum on 11 Dec 2006. [Credit: SOHO

data archive].

the Sun happened to go through a long-term minimum period of activity during1645-1715. Since the invention of telescope, daily sunspots have been observedfor nearly four hundred years. Sunspot number series (see later) is the most usedindex of solar activity and probably the most analyzed time series in astrophysics[Usoskin and Mursula, 2003].

With the development of solar telescope, the structure of sunspots could beobserved in detail. Most sunspots are observed to contain two parts known as theumbra and the penumbra. The umbra is at the center and the darkest part of asunspot and has the strongest and vertical magnetic field of the sunspot, while thepenumbra, surrounding the umbra, is a little lighter and its magnetic is weaker andnearly horizontal. Figure 2.2 depicts the detected structure of the sunspot groupshown in Fig. 2.1 observed by Hinode Satellite. Accordingly, the areas of theumbra and penumbra can be obtained. The continuous measurement of sunspotareas began at Greenwich Observatory in 1874.

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Fig. 2.2. The same sunspot group as shown in Fig. 2.1 observed with Hinode/SOT

G-band. Credit: Hinode Science Center/NAOJ.

2.1.1. Wolf sunspot numbers

The most widely used index of solar activity is the Zurich or Wolf sunspot number(now continued by, and even referred to as the International sunspot number),often expressed by Rz, RW , RI , R, WSN, SSN or SN, which was introduced byRudolf Wolf of Zurich Observatory in 1850s. Wolf started daily observations ofsunspots in 1848 soon after Heinrich Shwabe discovered the sunspot activity cyclewith a period of about 10 years in 1844 [Bray and Loughhead, 1964; Mundt et al.,1991]. Recognizing the difficulty in identifying individual spots and the importanceof sunspot groups, Wolf in 1848 defined his “relative” sunspot number index as

Rz = k(10G+N), (2.1)

where G is the number of sunspot groups, N the number of individual sunspots inall groups visible on the solar disc and k denotes the individual correction factorwhich compensates differences in observational techniques and instruments usedby different observers, thus normalizing the different observations to each other[Wilson, 1992].

Rudolf Wolf worked at Zurich Observatory as the “primary” observer (judgedas the most reliable observer during a given time) in 1848-1893. He extended theseries back to 1749 by using the observations from the primary and secondaryobservers and filled in the gaps by using linear interpolation and geomagnetic dec-lination measurements as proxies. Therefore, Wolf sunspot numbers are a mixtureof direct sunspot observations and calculated values [Hoyt and Schatten, 1998a].The primary observers for the Rz series were Staudacher (1749-1787), Flaugergues(1788-1825), Schwabe (1826-1847), Wolf (1848-1893), Wolfer (1893-1928), Brunner(1929-1944), Waldmeier (1945-1980) (at the Swiss Federal Observatory, Zurich),

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and Koeckelenbergh (since 1980, at the Royal Observatory of Belgium, Brussels).If the observations from the primary observer are not available for a certain day,then the observations from the secondary, tertiary observer, etc. are used. Us-ing the observations from a single observer aims to make Rz a homogeneous timeseries.

The Rz index was provided and maintained by the Swiss Federal Observatory inZurich, Switzerland through 1980. Since 1981 to the present the Rz index has beenprovided and maintained by the Sunspot Index Data Center in Brussels, Belgium[Hathaway et al., 2002]. The data process was changed in 1981 from using theobservation of the primary observer to using a weighted average of observationsby many observers when the Royal Observatory of Belgium took over the processof preparing the International sunspot numbers from the Zurich observatory.

Yearly values of Rz series are available since 1700, monthly values start from1749 and daily values from 1818 [Friedli, 2005]. However, the data is reliableonly since Wolf’s time, i.e., 1848. The earlier data which was reconstructed byWolf has pretty large uncertainties due to serious data gaps and the differentcalculation techniques used by observers. Data gaps of a few days in 1818-1848are manageable. Data gaps of many months in 1749-1848 and of a few years in1700-1749 were filled by Wolf. There are many filled values in 1700-1817 resultingin a rather poor sunspot index during this period. Moreover, Wolf obtained thedaily values from his correspondents without checking the original observationshimself. Thus significant uncertainties exist in the definition of G and N due todifferent observers. Vitinsky et al. [1986] estimated the systematic uncertainties tobe about 25% in monthly sunspot numbers. The quality of the data is considered as‘hardly reliable’ before 1749, ‘questionable’ during 1749-1817, ‘good’ during 1818-1847, and ‘reliable’ from 1848 onwards [Eddy, 1976; McKinnon and Waldmeier,1987; Kane, 1999]. The Rz index provided the longest continuous measure of solaractivity [Waldmeier, 1961] until another sunspot index, Group sunspot number,was established in 1990s [Hoyt and Schatten, 1998a], which extends back to 1610.

Usually, the daily values are averaged to monthly values in order to removevariations associated with the Sun’s 27-day synodic rotation period. The monthlyvalues still have large oscillations and require additional averaging processes toproduce a signal which varies smoothly from month to month. A commonly usedaveraging method is the 13-month running mean of the monthly values. If Rn isthe monthly value for month n, then the 13-month running mean is given by

Rn =1

24(Rn−6 + 2

5∑i=−5

Rn+i +Rn+6), (2.2)

simply called the smoothed sunspot number [Hathaway et al., 1999]. Solar cyclemaxima and minima are usually determined by these smoothed monthly values.The period between two successive minima defines the solar cycle. Solar cycle 1started in March 1755 and the present cycle 24 started in January 2009 [Hathaway,2010].

Gleissberg filter/12221 filter is often used to smooth the values of solar cyclemaxima in order to suppress the noise when studying long-term variations of solar

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activity [Gleissberg, 1944; Soon et al., 1996; Mursula and Ulich, 1998]. The Gleiss-berg filtered solar maximum of cycle n [Gleissberg, 1944; Petrovay, 2010] has theform

(Rmax)nG =1

8(Rn−2max + 2Rn−1max + 2Rnmax + 2Rn+1

max +Rn+2max). (2.3)

2.1.2. Group sunspot numbers

Noting that no error estimates are available in the Rz series and some earlysunspot observations were not included in Wolf’s collection, Hoyt and Schatten[1998a;b] collected many more observations than Wolf did, and introduced theGroup sunspot numbers since the observations from a few early observers have onlythe daily number of sunspot groups, but not the number of individual sunspots.The daily Group sunspot number RG is defined as

RG =12.08

N

∑i

k′iGi (2.4)

where Gi is the number of sunspot groups recorded by the ith observer, k′i is theith observer’s individual correction factor, N is the number of observers used toform a daily value, and 12.08 is a normalization number scaling RG to Rz valuesin 1847-1976. The normalization number varies slightly due to the total numberof observations used for different time period.

The main importance of the RG series is in extending the sunspot number ob-servations back to the earliest telescopic observations in 1610. Hoyt and Schatten[1998b] compiled 455242 sunspot observations from 463 observers in 1610-1874.Only a few observers’ presentations are missing on Hoyt and Schatten [1998b]because their observations were burt or are missing. Much effort was taken to col-lect the observations before 1874, when the Royal Greenwich Observatory startedrecording detailed sunspot observations including their position and area. Thedatabase for RG has about 80% more daily observations than Rz for the period of1610-1874. Therefore, the RG series is more reliable and homogeneous than theRz series, especially before 1874. The two series agree closely with each other since1950s [Hoyt and Schatten, 1998b; Letfus, 1999]. The main characteristics of solarcycles obtained from the two time series also agree with each other [Hathaway etal., 2002].

The RG series includes not only one primary observation but all available obser-vations. This approach allows to estimate systematic uncertainties of the resultingRG. The systematic uncertainties for the yearly RG values are caused by fouraspects: missing observations; uncertainties in the values of k′; random errors inthe daily values; and drifts in the k′ values. The final systematic error is about10% before 1653, less than 5% in 1654-1727 and 1800-1849, 15-20% in 1728-1799,and about 1-2% in 1851-1995 with error rising mainly from missing observations[Hoyt and Schatten, 1998b]. The major result is that solar activity according toRG in 1700-1882 is 25-50% lower than that given by Wolf. Consequently RG sug-

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gests that solar activity in the last few decades has been higher than it was duringseveral earlier centuries.

The RG series provides a pure sunspot series without filling in missing monthsor years. The number of days used to form the monthly mean values is alsoavailable for the series. This allows users to evaluate the data coverage for eachperiod and to estimate related errors. The RG series has still some uncertaintiesand inhomogeneities [Letfus, 2000]. However, the great advantage of this seriescompared to Rz is that the uncertainties can be estimated and taken into account[Hoyt and Schatten, 1998b; Usoskin and Mursula, 2003].

Yearly RG values are available from 1610 to 1995. Within this 386-year inter-val there are 32 years when RG has an unreliable value (uncertainty greater than25%) or is missing. The RG series has not been continued beyond 1995. There areminor problems in the work of Hoyt and Schatten [1998b]. E.g., the Julian calen-dar dates were taken to be Gregorian in some cases, and individual sunspots wereoccasionally defined as sunspot groups [Vaquero, 2007]. Also some published ob-servations were not covered in this work. Recently a great amount of old drawingsof sunspot observations have been discovered and published [Vaquero, 2004; Arlt,2008; 2009a;b; 2011]. Therefore, improvements on the work of Hoyt and Schatten[1998b] can be obtained.

In addition to Rz and RG, there are some other sunspot numbers, such as theBoulder sunspot numbers and the American sunspot numbers. These additionalnumbers can be scaled to the International Sunspot Numbers by using appropriatecorrection factors [Hathaway, 2010].

2.1.3. Sunspot areas and positions

Another widely used solar activity index is the sunspot areas. In May, 1874 theRoyal Greenwich Observatory (RGO) started compiling sunspot observations froma small network of observatories including Greenwich, England; Cape Town, SouthAfrica; Kodaikanal, India; and Mauritius to produce daily sunspot positions andareas. Sunspot areas were measured in units of millionths of a solar hemisphere(µHem) by dividing the sunspot in photographs into small squares. Both umbralareas and whole spot areas were calculated. The measured areas were corrected forforeshortening on the visible disc. The accuracy of sunspot areas was determinedup to 0.1 µHem. Sunspot positions were measured relative to the center of solardisc, i.e., latitude and central meridian distance. The accuracy of sunspot positionswas determined to 0.1 degrees. The RGO stopped preparing its data in 1976 whenthe US Air Force (USAF) started compiling sunspot area and position data fromits Solar Optical Observing Network (SOON) in 1977 with the help of the USNational Oceanic and Atmospheric Administration (NOAA).

The sunspot areas reported by USAF/NOAA are derived from sunspot drawingsprepared at the SOON sites, which were initially measured by overlaying a gridand counting the number of cells that a sunspot covered and later (since late 1981)by employing an overlay with a number of circles and ellipses of different areas.

21

Sunspot positions are determined at the accuracy of 1 degree. Sunspot areasare determined by rounding to the nearest 10 µHem. The combined dataset ofRGO and USAF/NOAA SOON, from May 1874 to the present, is available onlineat http://solarscience.msfc.nasa.gov/greenwch.shtml. The average daily sunspotarea over each solar rotation is shown in the bottom panel of Fig. 4.1.

Wilson and Hathaway [2006] have studied the measurements taken in RGOand SOON in detail. They also studied the relationship between sunspot area andsunspot number among RGO, SOON, Rome Observatory, Swiss Federal Observa-tory and Royal Observatory of Belgium. The relationship between total correctedsunspot area and sunspot number is not good for daily values, but becomes bet-ter when using monthly or yearly averages. However, the sunspot areas reportedby USAF/NOAA are significantly smaller than those given by RGO [Fligge andSolanki, 1997; Baranyi et al., 2001; Hathaway et al., 2002]. Figure 2.3 obtained byHathaway [2010] shows the relationship between sunspot area and sunspot numberusing 13-month running mean values. Top panel shows that the smoothed spotarea reported by RGO and the smoothed International sunspot number have veryhigh correlation (r = 0.994, r2 = 0.988). The best fit to the two quantities givesa slope of 16.7 and an offset of about 4. The zero point for the sunspot number isshifted slightly from zero because a single sunspot does not give a sunspot numberof 1, but 11 (see E.q. 2.1) for a correction factor k = 1.0. Therefore, the rela-tionship between the RGO sunspot area ARGO and sunspot number Rz is roughlydescribed as

ARGO = 16.7Rz. (2.5)

Bottom panel shows the relationship between the smoothed USAF/NOAA SOONsunspot areas and smoothed Rz.The slope of the fitting line is 11.32. Accordingly,we have

ASOON = 11.32Rz. (2.6)

To keep the later USAF/NOAA SOON sunspot areas consistent with the earlierRGO sunspot areas, the USAF/NOAA SOON sunspot areas should be multipliedby 16.7/11.32 ≈1.48.

Digitizing the drawings of sunspot observations by Samuel Heinrich Schwabe,Arlt [2011] calculated sunspot positions and sizes in 1825-1867. Another largeseries of sunspot positions was obtained by Arlt [2008; 2009a] for the years 1749-1796 from the drawings of Johann Staudacher. A number of positions of sunspotswere determined by Arlt [2009b] from the observations at Armagh Observatory in1795-1797.

2.1.4. Cyclicities in solar activity

The most prominent periodicity of sunspot activity is the 11-year cycle apart fromthe Sun’s 27-day rotation period. The 11-year cycle is also called the Schwabecycle, discovered by Schwabe [1844] using a number of daily sunspot groups in1826-1843. This cycle dominates the sunspot activity during almost the whole

22

Fig. 2.3. The relationship between sunspot area and sunspot number obtained by Hath-

away [2010]. All the data used in this figure are 13-month running mean values. Top

panel: RGO sunspot area vs. the International sunspot number in 1997 - 2010. The

slope of the fit line is 16.7. Bottom panel: USAF/NOAA SOON sunspot area vs. the

International sunspot number in 1977 - 2007. The slope of the fit line is 11.3.

23

observed time interval. However, this cycle varies in amplitude, length and shape[Usoskin and Mursula, 2003].

The idea of regular variations in sunspot numbers was suggested by the Danishastronomer Christian Horrebow in 1770s on the basis of his sunspot observationsin 1761-1769 [Gleissberg, 1952; Vitinskii, 1965]. Unfortunately, his results wereforgotten and the data lost. Later, in 1843, the amateur astronomer HeinrichSchwabe found that sunspot activity varies cyclically with the period of about10 years [Schwabe, 1844]. This was the beginning of the continuing study ofcyclic variations of solar activity [Usoskin and Mursula, 2003]. Figure 2.4 showsthe Schwabe cycle in total monthly sunspot group numbers and in the standardsmoothed sunspot numbers. The 11-year Schwabe cycle is the most prominentcycle in the sunspot series and in most other solar activity parameters. In addition,many other parameters including heliospheric, geomagnetic, space weather, andclimate parameters also show the 11-yr periodicity.

The background for the 11-year Schwabe cycle is the 22-year Hale magneticpolarity cycle, which is also known as Hale’s polarity law. Hale [1908] found thatleading spots and trailing spots in a binary sunspot group have opposite magneticpolarity. And that leading spots in northern and southern hemispheres have alsoopposite magnetic polarity. Furthermore, the polarity of sunspot magnetic fieldchanges in both hemispheres when a new 11-year cycle starts, see Fig. 2.5. Thisrelates to the reversal of the global magnetic field of the Sun with the period of 22years. It is often considered that the 11-year Schwabe cycle is the modulus of thesign-alternating Hale cycle [Sonett, 1983; Bracewell, 1986; Kurths and Ruzmaikin,1990; de Meyer, 1998; Mininni et al., 2001]. Although the 22-year magnetic cycle isthe basis for the 11-year Schwabe cycle, a 22-year cycle is not expected in the total(unsigned) SSN if the dynamo process is symmetric with respect to the changingpolarity. Gnevyshev and Ohl [1948] studied the intensity (total number of sunspotsover a cycle) of solar cycles and showed that solar cycles are coupled in pairs of aless intensive even-numbered cycle followed by a more intensive odd cycle. Thisis called the Gnevyshev-Ohl (G-O) rule or even-odd rule. Using Rz, the G-O ruleworks only since cycle 10 and fails for cycle pairs 4-5, 8-9 and 22-23 [Gnevyshevand Ohl, 1948; Storini and Sykora, 1997]. When using RG, the G-O rule is validsince the Dalton minimum and, in the reverse order, even before that [Mursula etal., 2001].

Silverman [1992] analyzed the power spectrum of monthly SSN for the period1868 to 1990, and indicated the presence of a 33.3-yr peak. Attolini et al. [1990]carried out a careful power spectrum analysis of the historical aurora observed sincecirca 700 BC in the East (China, Japan, Korea) and in central Europe at latitudesbetween 30 and 45 and found a period of 66 years. The three-cycle quasi-periodicity also exists in a shallow water core taken from the Ionian Sea (GallipoliTerrace) [Castagnoli et al., 1997]. Ahluwalia [1998] analyzed the Ap index andsunspot numbers in the last 6 decades, and pointed out that there is a three-cyclequasi-periodicity in both data. Ahluwalia [2000] further indicated the existence ofthree-cycle quasi-periodicity in the interplanetary magnetic field intensity, and thesolar wind bulk velocity in 1963-1998. Currie [1973] found a period of 66.7-yr inmonthly sunspot numbers in 1749-1957 by using the maximum entropy method.

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Fig. 2.4. The 11-year Schwabe cycle seen in the monthly number of sunspot groups and

the daily number of sunspots [Li, 2009a]. Top panel: monthly total number of sunspot

groups in 1876 - 2009. Bottom panel: the monthly mean of daily sunspot numbers for

the same time period. The thick lines show the corresponding 13-month running mean

values.

Fig. 2.5. Hale’s polarity laws [Hathaway, 2010]. Left panel: A magnetogram from sunspot

cycle 22 (2 August 1989). Yellow color denotes positive polarity and blue denotes negative

polarity. Right panel: A corresponding magnetogram from sunspot cycle 23 (26 June

2000).

25

According to the relationship between SSN and Ap, he successfully predicted thesize of cycle 23 (119). However, all these results are criticized by Wilson andHathaway [1999], who claimed that they are mainly due to the short period andpoor quality of data.

The long-term variation of the amplitude of Schwabe cycles is known as theGleissberg cycle [Gleissberg, 1944]. However, the Gleissberg cycle is not a cyclein a strict sense but rather a modulation of the cycle envelope with a varyingtime scale of 50-140 years [Gleissberg, 1971; Kuklin, 1976; Ogurtsov et al., 2002].The period of about 200 (160-270) years is called Suess or de Vries cycle [Schove,1983; Ogurtsov et al., 2002]. Suess [1980] found a significant 203 year variationin tree-ring radiocarbon records. Even longer (super-secular) cycles are found incosmogenic isotope data. Most prominent cycles are the 600-700-yr cycle and the2000-2400-yr cycle [Usoskin et al., 2004]. Gleissberg [1952] also suggested a 1000-yrvariation.

2.1.5. Grand minima and maxima

The regular cyclicity of solar activity is sometimes intervened by periods of greatlydepressed activity called grand minima. The last grand minimum and the onlyone covered by direct solar observations was the famous Maunder minimum during1645-1715 [Eddy, 1976; 1983]. There are other grand minima found in the pastfrom cosmogenic isotope data in terrestrial archives like 14C and 10Be, includingSporer minimum (1420-1530), Wolf minimum (1280-1340), Oort minimum (1010-1050), Dalton minimum (1790-1830) etc [Eddy, 1983; Kremliovsky, 1994; Komitovand Kaftan, 2004]. During the Dalton minimum, however, sunspot activity wasnot completely suppressed and still showed the Schwabe cyclicity. Schussler et al.[1997] suggested that this can be a separate, intermediate state of the dynamobetween the grand minimum and normal activity.

Grand minima and maxima are an enigma for the solar dynamo theory. It isintensely debated what is the mode of the solar dynamo during such periods andwhat causes such minima and maxima [Feynman and Gabriel, 1990; Sokoloff andNesme-Ribes, 1994; Schmitt et al., 1996]. A large amount of work suggests thatthey are the result of random fluctuations of the dynamo governing parameters[Hoyng, 1993; Brandenburg and Sokoloff, 2002; Kowal et al., 2006; Moss et al.,2008; Usoskin et al., 2009].

Due to the recent development of precise technologies, such as the improvementin accelerator mass spectrometry, solar activity can be reconstructed over multiplemillennia from concentrations of cosmogenic isotopes. Accordingly, several grandminima and maxima in the past were found from there, e.g., Eddy [1977a]; Vosset al. [1996]; Goslar [2003]; McCracken et al. [2004]; Usoskin et al. [2006]; Brajsaet al. [2009]; Hanslmeier and Brajsa [2010]. Stuiver and Braziunas [1989] studiedthe 14C record and suggested two types of grand minima: shorter Maunder-typeand longer Sporer-like minima [Stuiver et al., 1991]. The production of cosmogenicisotopes in the Earth’s atmosphere is not only defined by solar activity, but also

26

Table 2.1. Approximate years (in -BC/AD) of the centers and the durations ofgrand minima in the long sunspot number series [Usoskin et al., 2007a].

No. year duration commentof center (years)

1 1680 80 Maunder2 1470 160 Sporer3 1305 70 Wolf4 1040 60 a)5 685 70 b)6 -360 60 a, b, c)7 -765 90 a, b, c)8 -1390 40 b, e)9 -2860 60 a, c)10 -3335 70 a, b, c)11 -3500 40 a, b, c)12 -3625 50 a, b)13 -3940 60 a, c)14 -4225 30 c)15 -4325 50 a, c)16 -5260 140 a, b)17 -5460 60 c)18 -5620 40 -19 -5710 20 c)20 -5985 30 a, c)21 -6215 30 c, d)22 -6400 80 a, c, d)23 -7035 50 a, c)24 -7305 30 c)25 -7515 150 a, c)26 -8215 110 -27 -9165 150 -

a) Discussed in Stuiver (1980) and Suiver & Braziunas (1989).b) Discussed in Eddy (1977a, b).c) Shown in Goslar (2003).d) Exact duration is uncertain.e) Does not appear in the short sunspot number series.

27

Table 2.2. Approximate years (in -BC/AD) of the centers and the durations ofgrand maxima in the long sunspot number series [Usoskin et al., 2007a].

No. year duration commentof center (years)

1 1960 80 modern, b)2 -445 40 -3 -1790 20 a)4 -2070 40 -5 -2240 20 b)6 -2520 20 a)7 -3145 30 a)8 -6125 20 -9 -6530 20 -10 -6740 100 -11 -6865 50 -12 -7215 30 -13 -7660 80 -14 -7780 20 -15 -7850 20 -16 -8030 50 -17 -8350 70 -18 -8915 190 -19 -9375 130 -

a) Discussed in Eddy (1977a, b).b) Center and duration of the modern maximum are preliminary since it is still ongoing.

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modulated by the geomagnetic field. The earlier reconstruction of solar activitydid not take the magnetic field into account.

Recently, Beer et al [2003] and Solanki et al. [2004] developed a more appro-priate reconstruction model by taking both effects into account [see also Usoskinet al, 2003; Usoskin et al., 2004]. Using this new model, the solar activity can bequantitively reconstructed. Accordingly, the grand minimum can be quantitivelydefined [Usoskin et al., 2006; 2007a] . Usoskin et al. [2007a] defined the grandminimum as a period when the sunspot number (smoothed with the Gleissbergfilter) is less than 15, while during grand maximum, it exceeds 50 during at leasttwo consecutive decades. Tables 2.1 and 2.2 show the grand minima and maximasince 9500 BC found by Usoskin et al. [2007a]. The new definition leads to longerdurations of grand minima. For example, the duration of Sporer minimum in Table2.1 is 50 years longer than that found by Eddy [1983].

Recent studies suggest that the Sun was more active in 1930s-2000 (cycles 17-23) than in other times during last thousand years. Accordingly, this time periodis called Modern Grand Maximum (MGM). However, more recent measurementshave shown that sunspot levels in the declining phase of cycle 23 and in the start-ing phase of cycle 24 are far below average [Livingston et al., 2012; Penn andLivingston, 2011]. The last minimum before present sunspot cycle 24 lasted muchlonger than earlier minima showing the Sun’s unusual low activity [Li, 2009a]. Themutual relation between sunspots and solar radiation indices, such as UV/EUVindices, the 10.7cm flux, has dramatically changed at the end of 2001 after remain-ing stable for about 25 years until 2000 [Lukianova and Mursula, 2011; Tappingand Valdes, 2011]. In addition, the average magnetic field strength in sunspotumbra in 1998-2011 has been decreasing by 46 G per year [Livingston et al., 2012].This implies that cycle 24 might mark the end of MGM.

2.2. Solar flares

A solar flare was first observed by Carrington and independently by Hodgson in1859 in white (visible) light as a localized brightening of small areas within acomplex sunspot group. It was also the first documented white light flare.

A solar flare is observationally defined as a brightening of any emissions of theelectromagnetic spectrum occurring at a time scale of minutes [Benz, 2008]. Solarflares are one of the two most important eruptive phenomena in the atmosphere ofthe Sun (the other are CMEs). They are the most efficient mechanism to releasemagnetic energy associated with active region filaments (prominences), sunspotsor sunspot groups. The total energy released by a large flare can reach 4·1025 J,but the average energy of a solar flare is 1023−1024 J. The energy released in solarflares can be distributed in many forms: hard electromagnetic radiation (γ- andX-rays), energetic particles (protons and electrons), and mass flow.

Small flares (microflares, nanoflares and X-ray bright points) can occur all overthe Sun, in active regions and in the magnetic network of quiet corona [Parker,1988; Benz and Krucker, 1998]. X-ray and EUV jets are frequently observed in

29

association with these small events [Shimojo et al., 1996]. The source regions ofsmall flares can be diagnosed by EUV and X-ray images. Large (or regular) flaresnormally occur in active regions with complex magnetic structures observed inphotospheric magnetograms. Flare prolific sites often show signatures of stronglytwisted or sheared magnetic fields. Only regular flares are discussed in the followingsections of this thesis.

2.2.1. Classification of solar flares

Fig. 2.6. Hα observation during a flare on 7 August 1972 by Big Bear Solar Observatory.

Before 1950s solar flares were mainly observed in Hα emission. Hα flares arealso called optical flares. They are classified to five classes denoted by S, 1, 2, 3,and 4, based on the area of Hα emission. The maximum brightness of Hα emissionis denoted by F, N, and B representing faint, normal, and bright. Accordingly, anSF flare belongs to the smallest class and has a faint brightness, while 4B denotesthe largest class and is bright. Figure 2.6 presents the solar atmosphere at theflare site in the wavelength band of the Hα line emission observed on 7 August1972 by Big Bear Solar Observatory. This particular flare, so-called seahorse flare,is an example of a two ribbon flare in which the flaring region appear as two brightlines threading through the area between two sunspots of a sunspot group. The

30

Fig. 2.7. GOES X-ray flux observed on 13-15 July 2000 during which an X-flare and

several M and C flares occurred.

Class W/m2 Ergs/cm2/sA I < 10−7 I < 10−4

B 10−7 ≤ I < 10−6 10−4 ≤ I < 10−3

C 10−6 ≤ I < 10−5 10−3 ≤ I < 10−2

M 10−5 ≤ I < 10−4 10−2 ≤ I < 10−1

X 10−4 ≤ I 10−1 ≤ I

Table 2.3. X-ray flare classification and peak intensity.

31

two ribbons are also connected to each other through bright Hα loops. A hot40-million-degree shell intersects the surface at the ribbons and condenses at theloop tops.

Recently, solar flares are usually characterized and classified by their brightnessin X-ray radiation since the continuous X-ray flux observations by GeostationaryOperational Environmental Satellites (GOES) started in mid-1970s. The X-rayflux is measured in two bands, 0.5-4.0 A and 1.0-8.0 A. X-ray flares are classifiedto A, B, C, M, and X classes representing increasing powers of ten (see Table 2.3).A-class and B-class are defined according to the X-ray flux in 0.5-4.0 A band,representing typical background levels around minimum and maximum of a solarcycle, respectively. C, M, and X classes are defined according to the X-ray fluxin 1.0-8.0 A band, indicating increasing levels of flaring activity. Accordingly, thebiggest flares are called X-class flares. Within these main classes the flares arefurther divided into subclasses. For instance, a C5 flare has the measured peakintensity of 5.0×10−6 W/m2 and an X3 flare 3.0×10−4 W/m2 . Figure 2.7 givesan example of GOES X-ray flux observation in three days by GOES 8 and GOES10. GOES 8 (1994-2004) and GOES 10 (1997-2009) both have 0.5-4.0 A bandand 1.0-8.0 A band. In Fig. 2.7 the X-ray flux observed by GOES 10 is overlaidby GOES 8 with perfect matching each other. A large impulsive increase on 14July 2000 was recorded in the 1.0-8.0 A band (red) with the value of 5.7×10−4

W/m2. Accordingly, the flare level was classified as X5.7. A few M-flares andmany C-flares were also observed in the three days. These C-, M- and X-flarescaused also the increases in the 0.5-4.0 A band (blue) but with lower intensity.

2.2.2. Phases of a typical flare

The flare process is mainly divided into four phases according to variation ofelectromagnetic radiation as presented in Fig. 2.8. At first, soft X-rays and EUVare observed to increase gradually due to the slow heating of coronal plasma inthe flare region. This period is called the pre-flare phase followed by the impulsivephase, in which most of the energy is released and a large number of energeticparticles (electrons and ions) is accelerated. An evident impulsive increase can beobserved in hard X-rays, γ-rays, and microwaves (millimeter to centimeter radiowaves), and a few impulses can be seen in decimetric and metric radio waves andEUV. A significant rapid increase is observed in soft X-rays. The fast increase inHα intensity and line width terms the flash phase, which coincides partly with theimpulsive phase. After the flash phase, plasma in the low corona (around the flareregion) returns nearly to its original state, while the plasma ejections in the highcorona, the magnetic reconfigurations, and shock waves keep accelerating particlescontinuously, causing metric radio bursts and interplanetary particle events. Thisperiod is named the decay phase. Noting that the fluxes of soft X-rays and Hαhave different temporal evolution, flare phases in soft X-rays are characterized asearly phase, impulsive phase, gradual phase and extended phase [Hudson, 2011].

In the past couple decades, many spacecraft (e.g., SMM, SOHO, TRACE,

32

time

Fig. 2.8. Phases of a typical flare (original figure by Benz [2008]).

Yohkoh, RHESSI, SORCE, Hinode, SDO, STEREO) have observed solar flaresin hard X-rays and γ-rays continuously with good spatial and temporal resolution.Many hard X-ray and γ-ray flares have been observed, even in weak flare classes.For example, 29 γ-ray flares were detected by RHESSI in 2002-2005, which cor-responded to GOES soft X-ray classes C9.6 to X17.0 [Ishikawa et al., 2011]. Thespace-based observations have provided many striking and excellent images. Oneimportant finding is that the coronal sources of hard X-rays and γ-rays are abovethe loop-top, showing systematically different signatures from the double footpointsources.

Krucker et al. [2008] present a good example of coronal hard X-ray sourcetogether with two footpoint X-ray sources as shown in the left panel in Fig. 2.9.This figure shows a TRACE 1600 A image taken at 06:45:11 UTC, 20 January2005, overplotted with 12-15 keV red contours (in the corona) and 250-500 keVblue contours at the two footpoints, implying that the coronal hard X-ray emissionsare fainter than the emissions of the footpoint sources. This made it so hard todetect the coronal source earlier. The high coronal hard X-ray and γ-ray emissionsshow a fast rise and a slower constant exponential decay and the corresponding

33

X (arcsecs)

Y (a

rcse

cs)

0020 0040

Fig. 2.9. Flare observations in hard X-ray and γ-rays. Left: two footpoint sources in γ-

rays and one coronal source in hard X-rays observed during 06:43 - 06:46 UT, 20 January

2005 [Krucker et al., 2008]. Right: flare observation from soft X-rays (GOES) to γ-rays

(RHESSI, in units of counts/s per detector) signifies three flare phases: the rise (or early)

phase, impulsive phase, and decay (or late) phase [Lin et al., 2003].

34

photon spectrum is harder (small value of power-law index) compared with theemissions from the footpoint sources. The right panel in Fig. 2.9 presents theRHESSI hard X-ray and γ-ray count rates for the 23 July 2002 flare together withGOES soft X-rays on the top [Lin et al., 2003]. The temporal evolution of hardX-rays and γ-rays determines three phases- rise phase, impulsive phase, and latephase. The impulsive phase overlap partly with the impulsive and gradual phasesin soft X-rays. During the rise phase the hard X-ray emission comes from thecoronal source appearing at the top of the flare loop. During the impulsive phasethe emission mainly comes from footpoints, while an additional coronal sourcebecomes visible in the late phase.

2.2.3. Radio emissions during solar flares

Among prominent characteristics of flares are the large number of line emissionsand the consequences of the rapid energy release to particles. For example, 331emission lines within the wavelength 3590-3990 A were observed in the 3B flare ofSeptember 19, 1979 [Fang et al., 1991]. In a flare electrons and nuclei are accel-erated nearly to the speed of light. They can produce electromagnetic radiationthroughout the spectrum from radio waves to X- and γ-rays, in which visible linesof Hα and Ca II H and K are among the most prominent emissions. Usually flaresproduce only emission lines, but large flares can produce continuum radiation (i.e.,white light radiation).

Energetic electrons emit radio emission in magnetized plasma. The radio burstsassociated with solar flares are classified in five types according to their dynamicalspectral features [Wild et al., 1963]. Type I bursts, the first observed radio bursts,are radio noise storms, composing of short-term (a few seconds) narrow band (a fewMHz) bursts superposed on an increased continuum. This type of bursts can lastfor hours to days caused by supra thermal electrons being accelerated repeatedlyin active regions. Radio noise storms can be observed in association with flares andCMEs. Top panel of Figure 2.10 shows a fraction of type I bursts [Payne-Scott,1949].

Type II and Type III are frequency drifting bursts (see bottom panel of Fig.2.10). Type II bursts drift significantly slower than type III and last for a longertime. The typical frequency drifts of Type II and Type III bursts are 0.25 and20 MHz/s, respectively. In normal cases, bursts drift from high to low frequenciesdue to the upward motion of electrons along magnetic field lines. The acceleratedelectrons emit radio emissions at the local plasma frequency and its harmonic.Reverse drifting of type III bursts was also observed due to the downward flow ofelectrons. Accordingly such bursts are called reverse type III, including ‘U’ shapeand ‘J’ shape [Pick and Vilmer, 2008]. Type II bursts occur a few minutes afterthe start of a flare, around the flash phase, and last for more than 10 min. TypeIII bursts start to be detectable already a few minutes before the flare impulsivephase.

Type IV bursts are radio continua from millimeter to dekameter wavelength

35

Fig. 2.10. Flare-related radio emissions. Top: type I radio bursts (original

figure from Payne-Scott [1949]). Bottom: idealized sketch of a complete radio

event in the progress of a flare (original figure from Typical HiRAS Observations,

http://sunbase.nict.go.jp/solar/denpa/hiras/types.html).

36

which can be observed after type II bursts in association with an intense flare.The associated flare is accordingly called type II+IV event. The probability oftype IV bursts causing a sudden geomagnetic storm is a function of their radioimportance (The radio importance of a flare is defined as the maximum flux densityat 10 cm multiplied by the duration).

Type V radio bursts are a broadband diffuse continuum with the highest fre-quencies at the start around 150 MHz. They last from tens of seconds to a fewminutes. In general, type V bursts follow type III and were suggested to be causedby emissions of relatively low energetic electrons at the harmonic of local plasmain a open magnetic field.

It should be noted that recent radio observations with high resolution show afew fine structures, e.g., the zebra pattern in type IV, spike bursts, and fiber bursts.Millisecond spikes were observed with the new solar broadband radio spectrom-eters of NAOC (National Astronomical Observatories, China). The brightnesstemperatures of spikes were suggested to be 1015 K and even higher. Moreover,submillimeter radio waves at a few hundred GHz caused by ultra relativistic elec-trons were also detected recently, together with hard X-rays and γ-rays [Raulin etal., 2004].

2.2.4. Flare models

A classical flare model is the ‘CSHKP’ (Carmichael-Sturrock-Hirayama-Kopp-Pneuman) model. In this model, a current sheet is produced above a closedmagnetic loop before or after magnetic field line reconnection, forming a invertedY type cusp-shaped magnetic field structure. Magnetic reconnection is the key-point in this model. Although this model has been extended by many authors, nosubstantial improvement on this model has been achieved so far. The magneticreconnection model was even questioned by Akasofu [1995]. However, with the ad-vent of space-borne X-ray image telescopes, more and more evidence for magneticreconnection have been obtained since 1990s.

Using X-ray images observed by Yohkoh, Shibata [1999] developed a plasmoid-induced magnetic reconnection model, in which the plasmoid ejection triggers theimpulsive phase. The sketch of this model presented by Shibata [1999] is shownin Fig. 2.11. The plasmoid (filament/prominence held by a twisted flux rope) issituated in the current sheet existing in the corona before a flare and prevents themagnetic field lines from reconnecting before a flare occurs. If the plasmoid startsto rise, the plasma surrounding the current sheet will fill in the region left behindthe plasmoid, forming an inflow and driving the sheared or anti-parallel magneticfield lines to reconnect and release energy very fast. Magnetic reconnection furtheraccelerates the plasmoid until the plasmoid is ejected away from the Sun. Whatdrives the plasmoid to move originally is not clear. It seems that the structure ofthis process is beyond our observing ability at the moment [Hudson, 2011].

A large amount of plasma in the reconnection region is accelerated to ener-getic (non-thermal) particles, electrons and ions. The downward ejected energetic

37

Fig. 2.11. A sketch of the plasmoid-induced-reconnection flare model [Shibata, 1999].

X (arcsecs)

Y (

arc

secs)

900 940 980

14

0 1

80

220

Fig. 2.12. HXR sources (left and middle panels are from Krucker et al. [2008]). Left: two

footpoint sources and one coronal HXR source at the SXR loop top shown by 18-22 kev

contours overlaid on SXR image observed with RHESSI during the 13 July 2005 flare

[Battaglia and Benz, 2006]. Middle: two footpoint sources and one coronal HXR source

above SXR loop top presented by 33-53 kev contours observed with Yohkoh during the

13 January 1992 flare [Masuda et al., 1994]. Right: cusp-shaped configuration observed

with Yohkoh/SXT during the same flare as shown in the middle panel [Shibata, 1999].

38

particles of the reconnection jet lose their energy through hard X-ray emission(bremsstrahlung) when they hit the top of the magnetic loop in the low corona,observed as the HXR loop top source in Fig. 2.11. The heated plasma at the looptop can be observed as downward flow in Hα in the transition region and chromo-sphere. The energetic particles continue propagating downward along the newlyreconnected magnetic field lines, emitting HXR at the footpoints of coronal mag-netic loop in the chromosphere where density is high enough for the high-energyparticles to have Coulomb collisions and slow down until reaching thermal speeds.In some cases, γ-rays and neutrons are observed resulting from the downwardpropagation of energetic ions from the accelerated site into the chromosphere.The heated plasma at the footpoints expands upwards along the magnetic fieldloop, filling up the existing coronal loop, observed as “evaporation” of chromo-spheric material in soft X-ray images. At the footpoints, enhanced Hα and EUVradiation is also observed. The two footpoints resulting from many reconnectedmagnetic lines form two bright ribbons. With the more and more magnetic fieldsreconnecting in the outer layers the two flare ribbons appear to move apart grad-ually.

The energy deposited in the upper chromosphere is transported into the lowerchromosphere and upper photosphere through thermal radiation and conduction.In some large flare cases, significant increase in white light continuum can beobserved from chromosphere and photosphere during the impulsive phase, and ishighly correlated with hard X-rays [Matthews et al., 2003]. It is pointed out thatwhite light should be detected from all flares if the sensitivity of telescopes is highenough [Hudson et al., 2006].

The motion of the plasmoid may cause waves and fast shocks, which can furtheraccelerate particles and cause solar energetic particle events (SEPs). The acceler-ation of SEPs is discussed more closely in section 2.4 since most SEP related flareshave accompanying CMEs and large SEPs are considered more likely to be accel-erated by the CME shocks [Grayson et al., 2009; Cliver and Ling, 2009; Hudson,2011].

It is generally accepted that flare energy is stored by sheared or anti-parallelmagnetic fields and released by magnetic reconnection. The above discussed mag-netic reconnection model describes the observed features of flares very well. Theupward streaming of energetic electrons is detected as Type III radio bursts drift-ing from high to low frequency due to the decreasing density of corona. ReverseType III bursts from downward streaming of electrons are occasionally detectedas well [Aschwanden et al., 1995]. The deduced acceleration sites of electrons lieabove the X-ray loops. Hard X-ray sources at the footpoints and the loop top havebeen observed by RHESSI in several events (Battaglia and Benz [2006]; see theleft panel of Fig. 2.12). A coronal hard X-ray source above the loop top before theimpulsive phase has been observed by Yohkoh hard X-ray images (Masuda et al.[1994]; middle panel of Fig. 2.12). A cusp-shaped configuration is well observedwith Yohkoh/SXT (Shibata [1999]; Tsuneta et al. [1992]; right panel of Fig. 2.12).An increasing number of radio, EUV and X-ray observations support the magneticreconnection model and the cusp geometry of flares.

However, to the date, most flare processes before the impulsive phase still re-

39

main unobservable. Some basic major questions have not been solved. A flareis generally considered as a coronal phenomenon with many chromospheric fea-tures observed when the energy is transported into the dense chromosphere. Butwhere the energy comes from and what triggers the magnetic reconnection is notvery clear yet. Observations of photospheric magnetic fields show that large-scalesheared magnetic fields are often detected in an active region before a flare, andan increase in the field strength is observed during some flare cases [Wang et al.,2004], suggesting a trigger mechanism due to the emergence of new magnetic fluxfrom below.

2.3. Coronal mass ejections

Fig. 2.13. A jet-like CME and a loop-like CME observed with SOHO/LASCO [Chen,

2011].

Coronal mass ejections are mass (plasma) and magnetic field eruptions witha large amount of plasma and magnetic fields expanding from the low coronaand ejected into the interplanetary space. They are the largest-scale eruptivephenomenon in the Sun. It is generally believed that the energy released in aCME originates from the free magnetic energy stored in the corona, i.e., non-potential magnetic energy in access of potential magnetic energy. This energy isreleased during magnetic reconnection into several energy forms, such as plasmaheating, particle acceleration, waves and kinetic energy of the ejecta.

CMEs are morphologically divided into two classes, narrow CMEs and normalCMEs, due to their angular width less or greater than ∼10. When a CME withan angular width of tens of degrees propagates from the Sun to the interplanetary

40

Fig. 2.14. Jet-like CME model [left panel; Chen, 2011] and loop-like CME model [right

panel; Forbes, 2000].

space along or close to the Sun-Earth line, it is observed to surround the wholesolar disc, i.e., the projection angle in the plane of sky recorded in a 2D image is∼360. This kind of CME is called the halo CME.

CMEs were grouped into impulsive and gradual types in 1980s-1990s accordingto the observed speed of evolution. It was suggested that impulsive CMEs concurwith solar flares, and gradual CMEs are correlated with eruptive filaments. Laterobservations show that CMEs, flares and eruptive filaments do occur together inmany cases, but no clear correlations are found. Therefore, CMEs can not besimply classified as impulsive or gradual and this classification of CMEs is nolonger in use.

With modern soft X-ray and EUV observations, structures of CMEs can beobserved in more detail, although many processes still remain unknown. Accord-ingly, CMEs are nowadays classified into jet-like and loop-like classes according tothe structure of magnetic fields (open or connected to solar surface as a loop). Itis generally agreed that a jet-like CME (see the left panel of Fig. 2.13) is causedby the magnetic reconnection within an emerging flux or between a coronal loopand the open magnetic fields of a coronal hole, resulting in a compact flare (see theleft panel of Fig. 2.14). This reconnection model is similar to the model of X-rayjets produced during microflares [Shimojo et al., 1996]. Most of narrow flares showjet-like eruption.

Loop-like CMEs normally consist of three parts, a closed frontal loop followedby a dark cavity and a bright core (see the right panel of Fig. 2.13). A sketchof the corresponding model is presented in the right panel of Fig. 2.14 [Forbes,2000]. Statistics show that very powerful flares above class X2.0 have a one-to-one

41

association with CMEs [Hudson, 2011; Yashiro et al., 2006]. Based on the abovediscussed model of solar flares, the present CME model has been developed (seeFig. 2.14), involving flaring X-ray loops, eruptive filament/prominence formingthe bright core in the cavity, and frontal loop.

The global picture of a loop-like CME is as follows. A twisted or sheared coronalmagnetic flux rope (similar to but normally larger than the flux rope in Fig. 2.11),perhaps holding a prominence, rises for some reason, driving the magnetic fieldlines under it to reconnect. The fast magnetic reconnection causes a flare in thelow corona close to solar surface. Meanwhile, the field lines tying to solar surfaceand keeping the flux rope from flowing away from the Sun are cut off duringthe reconnection. When the flux rope moves outwards very fast, it stretches themagnetic field lines lying above it to form the frontal loop and causes a shockwave as well, which can further accelerate particles in the interplanetary space.The stretching or expanding frontal loop leads to low plasma density behind itwhich is seen as a dark cavity in white-light.

Unlike flares, the magnetic reconnection is not always necessary in the eruptionof flux rope during a CME. Many other processes can trigger the flux rope to erupt,such as loss of equilibrium and MHD instability [Vrsnak et al., 1991; Vrsnak, 2008].

2.4. Solar energetic particles acceleration

Solar energetic particles (SEPs), mainly protons, and a smaller amount of electronsand heavy ions, were first noted by Forbush as three unusual cosmic ray increases[Forbush, 1946]. Later they were detected by ground neutron monitors and diag-nosed as ground level events (GLEs; also called ground level enhancements). Onlyextreme high-energy particles (∼1GeV) from powerful solar eruptions can cause aGLE. Statistics show that SEPs occur every four days on an average, while GLEsoccur only once a year [Duggal and Pomerantz, 1971]. GLEs are suggested to beassociated with the impulsive phase of flares [Aschwanden, 2012], while large SEPsare proposed to be mainly accelerated by CME shocks [Cliver and Ling, 2009].

After a few decades of observations and studies, the SEP events were dividedinto two types, impulsive events and gradual (long-duration) events according tothe duration of soft X-rays [Cane et al., 1986; Reames, 1999]. The impulsive SEPswere thought to relate with flare acceleration processes and have accompanyingType III radio emissions, high ratio of Fe/O, enhanced 3He and high charge stateof Fe. The impulsive SEPs probably result from wave-particle interaction causedby magnetic reconnection during a flare, as mentioned in Sect. 2.2.4. The gradualSEPs were associcated with CMEs and Type II radio burst events, being acceler-ated by coronal and interplanetary shocks driven by CMEs. Gradual events have alower Fe/O ratio and a lower Fe charge state than impulsive events [Cliver, 2009].

However, recent observations show that ‘impulsive’ characteristics are some-times observed also during gradual events [Cane et al., 2006]. Considering thenewly observed signatures of SEPs and the suggested acceleration models, Cliver[2009] presented a sketch of a new SEP acceleration model (see Fig. 2.15) and

42

Fig. 2.15. A sketch of a new SEPs acceleration model [Cliver, 2009].

suggested a revised classification of SEP events [see Table 2 of Cliver, 2009, fordetails]. According to this scheme, SEP events are classified into flare and shockSEP classes instead of impulsive and gradual classes. Shock class has two sub-classes, quasi-perpendicular shocks and quasi-parallel shocks (denoted by Q-perpand Q-par, respectively in Fig. 2.15). Large (gradual) SEP events with impulsivecharacteristics are considered to result from quasi-perpendicular shock acceleratedsupra-thermal particles originating from flares. While large SEP events withoutimpulsive characteristics are quasi-parallel shock accelerated supra-thermal parti-cles originating from CMEs and solar winds [Cliver and Ling, 2009].

3. Solar rotation

3.1. Heliographic coordinates

In order to determine a position on the Sun or in the heliosphere we have toemploy a coordinate reference frame. For this purpose we first need to know therotation axis of the Sun. The direction of the rotation axis of the Sun is givenby two rotation elements or angles. One is the angle of inclination i between theecliptic plane and the solar equatorial plane. The other is the ecliptic longitude ofthe ascending node of the Sun’s equator determined by the angle, in the eclipticplane, between the vernal equinox direction and the ascending node, where theSun moves to celestial north through the ecliptic plane.

Once the axis of solar rotation is known, the solar latitude ψ, also called theheliographic latitude, is easily defined as the angular distance from the equator.Frequently used is also the polar angle (angular distance to the North Pole), orthe so called co-latitude θ=π/2-ψ.

However, there are no fixed points on solar surface to define the solar longitude.Defining the longitude is even more difficult for a differentially rotating surfacelike the Sun. Carrington introduced in 1850s a rigidly rotating (14.1844 deg/day,sidereal) longitudinal frame by observing low latitudinal sunspots. This is therotating rate of sunspots close to the latitude of 17-20. The sidereal rotating pe-riod of this longitudinal coordinate frame is 25.3800 days and its synodic period is27.2753 days, which is the so-called Carrington rotation (CR) period. CR number1 was defined to begin at 21:38:44 UT on 9 November 1853 with longitude 0 atthe central meridian of solar disc and increasing westwards to 360. CR number2105 started on the Christmas Eve in 2010. At the start of each new rotation thelongitude value of φ =0 is given to the center of the solar disc.

3.2. Surface rotation

The Sun’s surface rotation was noted by Harriot, Fabricius, Galileo, and Scheinerin the early decades of the 17th century, soon after the invention of telescope by ob-

44

serving the sunspots across the solar disc. Harriot did not publish his observationsof sunspots although he shared them with a group of correspondents in England[The Galileo Project: Thomas Harriot 1560-1621]. Johannes Fabricius publisheda book in 1611 named ‘Narration on spots observed on the Sun and their appar-ent rotation with the Sun’. Unfortunately, the existence of the book cannot befirmly verified up to the time. Scheiner and Galileo independently published theirobservations in 1612. Galileo gave a rotation period of 29.5 days. With improve-ment of the telescope, Scheiner obtained a rotation period of 27 to 28 days fromthe observations in 1609-1625 [Scheiner, 1630]. Scheiner also noted the differentialrotation on the solar surface. His book ‘Rosa Ursina’, published in 1630 presentedevidence that the pass of sunspots across the solar disc took a slightly longer timeat high latitudes than near the equator. However, the heliographic latitude andlongitude reference frame was not established until 1863 by Carrington.

It took 250 years until Carington established, by tracking sunspots, that thesurface solar rotation rate decreases with increasing heliospheric latitude, implyingthat the Sun’s outer layers are in a state of differential rotation. He determined the(sidereal) surface rotation rate (in degrees per day) as a function of heliographiclatitude to be as follows

Ω(ψ) = 14.25− 2.75sin7/4ψ. (3.1)

However, for practical purposes the power 7/4 is quite awkward. A more modernapproach is to expand the rotation rate as

Ω(ψ) = A+Bsin2ψ + Csin4ψ, (3.2)

where A is the equatorial rotation rate. A, B, and C need to be determined fromobservation.

In addition to sunspots, many other tracers, e.g., photospheric plasma (Dopp-lergrams), coronal holes (EUV intensity maps) and surface magnetic fields (mag-netograms) have been used to determine the rotation rate of solar surface. Someexamples of values for A, B and C determined from different tracers are shown intable 3.1. From table 3.1 one can see that: (1) the rotation parameters obtainedfor different tracers are not necessarily in agreement with each other; (2) the twosets of parameters determined by all sunspots are close to each other and so arethe groups and recurrent spots; (3) the sunspot groups and recurrent spots havea lower value of A than sunspots on an average, indicating that these have beenbraked by the slower rotation of the solar plasma; (4) the value of B is exception-ally small for coronal holes, which implies that coronal holes rotate quite rigidly.(More recent results on solar surface differential rotation will be discussed in Sect.5.4.2)

45

authors tracers A B CHoward et al. [1984] Mt. Wilson; all spots 14.522 -2.840

1921-1984Balthasar et al. [1986] Greenwich; all spots 14.551 -2.870

1874-1976 14.545 -2.720 -0.580Howard et al. [1984] Mt. Wilson; sunspot groups 14.393 -2.946

1921-1984Newton and Nunn [1951] Greenwich; recurrent spots 14.368 -2.690

1878-1944Scherrer et al. [1980] Wilcox; photospheric plasma 14.440 -1.980 -1.980

1976-1979Snodgrass [1983] Mt. Wilson; surface magnetic 14.366 -2.297 -1.624

fields 1967-1982Wagner [1975] OSO-7; coronal holes 14.296 -0.510

May 1972-Oct. 1973

Table 3.1. Coefficients of Eq. 3.2 describing the differential rotation of the Sun’ssurface (in deg/day) measured from different tracers.

3.3. Internal rotation

Helioseismology has revolutionized the study of differential rotation since we cannow empirically determine the rotation even inside the Sun. By using helioseismol-ogy inversion of 5-min oscillation data, GONG (The Global Oscillation NetworkGroup) found out that the rotation velocity at the poles is 3/4 of that at theequator (see Fig. 3.1). Differential rotation continues throughout the convectionzone. However, near the base of convection zone, rotation changes its characterand becomes solid-body rotation, which continues in the radiative interior. Thisregion of sudden change in the rotation rate is the so-called “tachocline” [Basuand Antia, 2003].

Basu and Antia [2003] have drawn a contour diagram of solar rotation as afunction of depth (see Fig. 3.2) by using solar oscillation data of GONG. FromFigs. 3.1 and 3.2 one can see that angular velocity does not change with latitude ordepth in the underlying radiative core, where the radius is less than 0.68R. Thisregion rotates almost rigidly and the rate is close to the rate of the convection zoneat 30 latitude. The rotation rate of convection zone at some latitudes, such as at15 and 60, does not change with depth, i.e., ∂Ω/∂r ≈ 0. There are two shearsin the rotation rate profile (Fig. 3.1), where the rotation rate increases rapidlywith the decreasing radius, i.e., ∂Ω/∂r < 0. One is close to the solar surface, fromthe surface to around 0.95R. The other appears at high latitude (above 30)around 0.8-0.7R, close to the tachocline. The shear close to the tachocline regionis generally believed to be responsible for driving the solar dynamo. The rotationrate decreases with decreasing radius at latitudes below 30.

According to the helioseismic solution of Charbonneau et al. [1999], Dikpati and

46

/r R⊙

Fig. 3.1. The solar internal rotation law (picture is taken from Elstner and Korhonen

[2005]).

Charbonneau [1999] suggested the following expression for the differential rotationΩ(r, ψ):

Ω(r, ψ) = Ωc +1

2[1 + erf(2

r − rcd

)]Ω(ψ)− Ωc, (3.3)

where Ω(ψ) = A+Bsin2ψ+Csin4ψ (Eq. 3.2) is the surface latitudinal differentialrotation and erf(x) is the error function. The best fit values of parameters are setas

Ωc/2π = 432.8 nHz (Ωc ' 13.46 deg/day),A/2π = 460.7 nHz (A ' 14.33 deg/day),B/2π = -62.69 nHz (B ' -1.95 deg/day),C/2π = -67.13 nHz (C ' -2.088 deg/day),rc = 0.71R,d = 0.05R.

The parameter rc denotes the radius of the tachocline center and d refers to thethickness of the tachocline. This definition characterized the differential rotationprofile of solar-like stars by purely latitudinal differential rotation Ω(ψ) with equa-torial acceleration, smoothly matching across a tachocline to a rigidly rotating“core” at a rate Ωc. This profile has been widely used in solar dynamo models.

47

r / R⊙

r / R⊙Fig. 3.2. A contour diagram of solar rotation as a function of equatorial (horizontal) and

polar (vertical) depth. The contours are drawn at intervals of 5 nHz. The thick contour

around the equator and close to the surface denotes Ω = 460 nHz [Basu and Antia, 2003].

48

Fig. 3.3. Residuals of the rotation rates of zonal flows at a few typical depths, obtained

using 2D regularized least-squares inversion of GONG data during 1995-2005 [Antia,

2006].

3.4. Torsional oscillation

Torsional oscillation is called the pattern of alternating bands (or zones) of faster(slower) than average rotation moving from mid-latitudes during cycle minimumtowards the equator (poles, respectively). During cycle maximum faster (slower)bands exist on the equatorial (poleward) side of mean latitude of activity. It isbelieved that this pattern extends throughout most of the convection zone.

The migrating zonal flow pattern was first detected by Howard and Labonte[1980] in Doppler measurements (digital full-disk velocity data from the 5250 Aline of Fe I) at the solar surface carried out at Mt. Wilson Observatory in 1968-1979. They found belts of faster than average rotation that migrate from mid-latitudes to the equator. The migration of the zonal flow bands during the solarcycle is closely connected to the migration of the magnetic activity belt. Theflow pattern was first detected helioseismically by Kosovichev and Schou [1997] inearly f -mode data from the MDI (Michelson Doppler Imager) aboard the SOHO

49

Fig. 3.4. Residuals of the rotation rates of zonal flows shown as a function of latitude at

the depth of r = 0.98R at a few selected times in 1995 - 2000. The thin solid line is for

GONG 1-3 months centered around June 1995; the light-dotted line is for June 1996; the

short-dashed line is for 1997 June; the long-dashed line is for June 1998; the thick solid

line is for June 1999; and the thick-dotted line is for June 2000 [Antia and Basu, 2000].

50

(Solar and Heliospheric Observatory) spacecraft, and was seen to migrate [Schou,1999]. Antia and Basu [2000] found that in addition to the pattern of convergingto equator with increasing time, there is another system of zonal flow bands athigh latitudes (>45) that migrate towards the poles as the solar cycle progresses.Figure 3.3 [Antia, 2006] shows the velocity residuals of zonal flows at a few repre-sentative depths and latitudes obtained using a two-dimensional (2D) regularizedleast-squares helioseismic inversion of GONG data in 1995-2005. Moreover, theamplitudes of the zonal flows at a certain depth change with latitude. Figure3.4 [Antia and Basu, 2000] presents residuals of the rotation rates of zonal flowsshown as a function of latitude at the depth of r = 0.98R at a few selected times.Considering the length of a solar cycle as 11 years [Vorontsov et al., 2002], thetime-varying rotation profile of zonal flows can be described as

Ω(r, θ, t) = Ω0(r, θ)+A1(r, θ)cos2πνt+A2(r, θ)sin2πνt ≡ Ω0(r, θ)+Asin(2πνt+φ),(3.4)

where Ω0 is the time-invariant part of the profile, t is time in years, with ν fixedat 1/11 cycles per year, and φ is phase (0≤ φ ≤2π).

Early results have shown that the zonal flow pattern can only be found fromthe surface [Howard and Labonte, 1980; Ulrich et al., 1988; Schou, 1999] to around0.9R [Howe et al., 2000a; Antia and Basu, 2000], not continuous to the base ofconvection zone or tachocline. With accumulation of more seismic data it has nowbecome clear that the zonal flow pattern penetrates though most of the convectionzone [Vorontsov et al., 2002; Basu and Antia, 2003; Howe et al., 2005; Antia etal., 2008]. Recently, Howe et al. [2009] made the first study of the differencesin the zonal-flow pattern between the cycle 22/23 minimum and the cycle 23/24minimum. They suggested that the extended cycle 23/24 minimum during thelast few years is probably due to the fact that the flow bands were moving moreslowly towards the equator. Antia and Basu [2010] confirmed this difference ofzonal-flow pattern between the two minima [Nandy et al., 2011]. They also foundthat the low-latitude prograde band that bifurcated in 2005 merged well beforethe cycle 23/24 minimum, which is in contrast to this band’s behavior during thecycle 22/23 minimum where the merging happened during the minimum. Thelength of solar cycle 23 in zonal pattern determined to be 11.7 years by correlatingzonal-flow pattern with itself, i.e., shorter than the length derived from sunspotnumbers (about 12.6 years; Hathaway [2010]). Howe et al. [2000b; 2007] reporteda 1.3-yr oscillation in the equatorial region around r = 0.72R during the risingphase of the solar cycle, but that has not been confirmed by other studies.

The origin of torsional oscillations or alternating zonal flows is not fully under-stood, but since the zonal flow banded structure and the magnetic activity patternseem to be closely related [Howe et al., 2000a; Antia et al., 2008; Sivaraman et al.,2008], it is generally believed that zonal flows arise from a nonlinear interactionbetween magnetic fields and differential rotation [Kuker et al., 1996; Covas et al.,2000; 2004].

51

3.5. Meridional circulation

Meridional circulation means a circular flow throughout the meridional plane in-cluding longitudinally averaged north-south and radial velocity components. Un-like differential rotation, meridional circulation is not yet fully constrained byobservation, but a poleward surface flow with the velocity of 10-20 m/s has beendetected by various techniques, such as Doppler measurements [Duvall, 1979; Hath-away, 1996], magnetic tracers [Komm et al., 1993; Hathaway and Rightmire, 2010],and helioseismic analysis [Giles et al., 1997; Braun and Fan, 1998]. The equator-ward return flow below the solar surface has not been observed yet. But it mustexist, since mass does not pile up at the solar poles [Dikpati et al., 2004].

Fig. 3.5. Solar meridional flow model. Left: Streamlines for the meridional circulation

denoted as small and big dots plotted at 1-yr time intervals. The meridional flow follows

along the streamlines. The flow is counterclockwise when observed near the surface. The

base of the meridional flow is theoretically constructed at r = 0.71R according to mass

conservation, right above the center of the tacholine ∼0.70R. Right: Typical meridional

flow pattern in the two hemispheres with streamlines denoted as arrows. [Dikpati et al.,

2004]

52

Meridional flows were initially found at the solar surface in 1970s [Duvall, 1979;Howard, 1979; Topka and Moore, 1982]. Howard et al. [1980] showed that thelarge-scale velocity patterns on the Sun include not only solar rotation but alsoanother component. This velocity field which they called the “ears” was found tobe symmetric about the central meridian. Beckers [1978] pointed out that the earshave a similar form to that expected from a meridional flow velocity. Subsequentwork [Labonte and Howard, 1982; Wohl and Brajsa, 2001] confirmed the merid-ional flows and found radial flows as well. van Ballegooijen and Choudhuri [1988]studied a model of meridional circulation and suggested a meridional circular flowvelocity form

ur(r, θ) = u0(Rr

)2(− 1

n+ 1+

c12n+ 1

ξn− c22n+ k + 1

ξn+k)ξsinmθ[(m+2)cos2θ−sin2θ],

(3.5)

uθ(r, θ) = u0(Rr

)3(−1 + c1ξn − c2ξn+k)sinm+1θcosθ, (3.6)

where ξ(r) = (Rr ) − 1, u0 is a measure of the velocity amplitude, and k and mare two free parameters which describe the radial and latitudinal dependence ofthe velocity, respectively (k > 0 and m ≥ 0). The constants c1 and c2 are chosenso that ur and uθ vanish at the base of the convection zone:

c1 =(2n+ 1)(n+ k)

(n+ 1)kξ−n0 , (3.7)

c2 =(2n+ k + 1)n

(n+ 1)kξ−(n+k)0 , (3.8)

where ξ0 = (Rr0 )−1. Applying this mathematical form to a flux transport dynamomodel Dikpati et al. [2004] obtained the typical meridional circulation flow patterndepicted in Fig. 3.5. The obtained maximum speed of the meridional flow closeto the surface is around 15 m/s.

However, because the meridional-flow speed is more than an order of magnitudesmaller than other flows at the solar surface, it is difficult to be measured accu-rately. Only local helioseismic techniques, e.g., the time-distance technique [Hill,1988] and ring-diagram technique [Duvall et al., 1993] can be used for such ananalysis. With improved observational techniques, the meridional flow has beenreliably measured at the solar surface using both magnetic elements as tracers[Komm et al., 1993; Hathaway and Rightmire, 2010] and Doppler velocities [Hath-away, 1996]. These observations show a general flow towards the poles in bothhemispheres that is roughly antisymmetric about the equator. The amplitude ofthis flow shows a variation over the solar cycle with fast flows (11.5 m/s) at mini-mum and slow flows (8.5 m/s) at maximum of cycle 23 [Hathaway and Rightmire,2010].

Recent results [Beck et al, 2002; Gonzalez Hernandez et al, 2008; 2010; Basu andAntia, 2010] imply that the pattern of the meridional flow also shows bands of fastand slow flows, very similar to those of the zonal flows and to the butterfly diagram,

53

Fig. 3.6. Antisymmetric pattern of meridional flows in northern and southern hemispheres

plotted as a function of time and latitude at the depth of r = 0.998R. Top: the speed

of meridional flows is overplotted with the position of sunspots. Bottom: the flow speed

is overplotted with zonal flow speed contours obtained by Antia et al. [2008]. To make

the plot symmetric about the equator, the sign of the flow has been reversed in the

southern hemisphere. Accordingly, positive values represent the flow towards the poles

in both hemispheres. The contours are spaced at intervals of 1 m/s with solid representing

prograde speeds and dashed representing retrograde [Basu and Antia, 2010].

54

as can be seen in Fig. 3.6. There is a good match between the positions of thesunspots and the region of low-speed meridional flows and between the migratingpattern of the meridional flow with that of the zonal flow. Basu and Antia [2010]found that the meridional flows not only vary with depth, and latitude, but thatthe time dependence of the variation is a function of both depth and latitude.Therefore, the flow form can be simply expressed as

u(r, θ, t) = u0(r, θ) + u1(r, θ)sin(ωt) + u2(r, θ)cos(ωt), (3.9)

where ω = 2π/T is the frequency of the solar cycle.Meridional flows play a key role in the operation of solar dynamo. They are

believed to be responsible for transporting magnetic elements towards the poles onthe solar surface [Wang et al., 1989] and towards the equator near the base of theconvective zone [Choudhuri et al., 1995; Dikpati and Charbonneau, 1999]. Theyalso play a role in the generation of differential rotation and its temporal variation[Glatzmaier and Gilman, 1982; Spruit, 2003] and in the spatiotemporal distribu-tion of sunspots and the solar cycle period [Choudhuri et al., 2007; Nandy andChoudhuri, 2002; Hathaway et al., 2003]. Variations in the equatorward merid-ional flow during a solar cycle have been suggested to be able to cause a deepsolar minimum [Nandy et al., 2011]. In the surface flux transport dynamos, theprolonged cycle 23/24 minimum can be explained by faster meridional flow andthe observed weaker polar fields compared to cycle 22/23 minimum [Schrijver andLiu, 2008; Wang et al., 2005; 2009], but the prolonged minimum conflicts withthe flux transport dynamos which produce stronger polar fields and a short cyclewith fast meridional flow [Dikpati et al., 2006; Choudhuri et al., 2007]. A fastmeridional flow in the surface flux models can slow down the process of oppositepolarities canceling each other across the equator. Therefore, the flow carries ex-cess of elements of both polarities to the poles with a only slight excess of elementswith the polarity of the polarward side of the sunspot latitudes. This requires alonger time to reverse the old polar fields and builds up weaker polar fields of theopposite polarity.

4. Active latitudes

Sunspots are known to appear preferably in a narrow high latitude belt at thebeginning of a solar cycle and approach the equator as the solar cycle progresses.Individual sunspot cycles can be separated near the time of minimum by the lat-itude of the emerging sunspots and, more recently, by the magnetic polarity ofsunspots as well. Sunspots of the old cycle and the new cycle coexist during afew years around solar minima. The latitude migration and the partial overlapof the successive sunspot cycles leads to the famous pattern of sunspots calledSporer’s law of zones by Maunder [1903] and was illustrated by his famous but-terfly diagram [Maunder, 1904]. NASA’s Marshall Space Flight Center updatestheir butterfly diagram together with average daily sunspot area every month athttp://solarscience.msfc.nasa.gov/SunspotCycle.shtml, which shows the latitudi-nal positions of sunspots for each rotation since 1874. Figure 4.1 shows an exampleof their butterfly diagram for the years 1874-2011 and the corresponding sunspotareas for the same time interval.

Hathaway et al. [2003] investigated the latitude positions of sunspot areas asfunction of time relative to the time of minimum.The centroid positions of sunspotareas in each hemisphere are calculated for each solar rotation. They also studiedthe area weighted averages of these positions in 6-month intervals and found thatall cycles start with sunspot zones centered at about 25 from the equator. Theequatorward drift is more rapid early in the cycle, slows down later in the cycleand eventually stops at about 7 from the equator [Hathaway, 2010].

Sunspots appear very often in pairs. The trailing sunspot of the pair tendsto appear further out from the equator than the leading spot. Thus the lineconnecting the trailing sunspot with the leading one is not parallel to the equator.Normally, the higher the sunspot latitude is, the greater is the inclination of theconnecting line to the equator. This pattern of sunspot emergence is called Joy’slaw.

While sunspots drift slowly to the equator, the magnetic fields at higher latitude(related to trailing sunspot polarities according to Joy’s law) are transported to-wards the poles where they eventually reverse the polar field at the time of sunspotcycle maximum. The radial magnetic field averaged over longitude for each solarrotation is shown in Fig 4.2 [Hathaway, 2010]. This magnetic butterfly diagram

56

Fig. 4.1. Top: A butterfly diagram of the latitude of sunspot occurrence versus time.

Bottom: average daily sunspot areas of each Carrington rotation in units of hundredths

of a solar hemisphere (10µHem). [Courtesy of Hathaway/NASA/MSFC.]

Fig. 4.2. Magnetic butterfly diagram constructed from the average radial magnetic field

of each Carrington rotation using the synoptic magnetic maps observed by Kitt Peak and

SOHO [Hathaway, 2010].

57

exhibits Hale’s law, Joy’s law, the polar field reversals as well as the transport ofhigher latitude magnetic field elements towards the poles.

5. Active longitudes

5.1. History of active longitude research

Solar activity occurs preferentially at some longitude ranges rather than uniformlyat all heliographic longitudes. These longitude ranges are called active longitudes.It has been found that various manifestations of solar activity, such as sunspots[Vitinskij, 1969; Bumba and Howard, 1969; Balthasar and Schussler, 1983; 1984;Berdyugina and Usoskin, 2003; Juckett, 2006], radio emission bursts [Haurwitz,1968], faculae and plages [Maunder, 1920; Mikhailutsa and Makarova, 1994], flares[Jetsu et al., 1997; Bai, 1987a;b; 1988; 1994; 2003a; Temmer et al., 2004; 2006a],prominences [Fenyi, 1923], coronal emission [Benevolenskaya, 2002; Badalyan et al.,2004; Sattarov et al., 2005], surface magnetic fields [Benevolenskaya et al., 1999;Bumba et al., 2000], heliographic magnetic field [Ruzmaikin et al., 2001; Takaloand Mursula, 2002; Mursula and Hiltula, 2004], total solar irradiance [Mordvi-nov and Willson, 2003], EUV radiation [Benevolenskaya et al., 2002], and sur-face helicity [Bao and Zhang, 1998; Pevtsov and Canfield, 1999; Kuzanyan et al.,2000] are non-axisymmetric and mainly occur in preferred longitude bands. Thesepreferred active longitudes were given different names when they were found indifferent indices, e.g., active sources [Bumba and Howard, 1969], Bartels activelongitudes [Bumba and Obridko, 1969], streets of activity [Stanek, 1972], sunspotnests [Casenmiller et al., 1986], hot spots [Bai, 1990], helicity nests [Pevtsov andCanfield, 1999], and complexes of activity [Benevolenskaya et al., 1999].

Even in the very early days of solar physics Carrington [1863] suspected thatsunspots do not form randomly over solar longitude but appear in certain longi-tudinal zones of enhanced solar activity. However, studies on this topic have beencarried out since 1930s [Chidambara Aiyar, 1932; Losh, 1939; Lopez Arroyo, 1961;Bumba and Howard, 1965; Warwick, 1954; 1965; Balthasar and Schussler, 1984;Wilkinson, 1991]. As a result, the tendency for active regions to cluster in pre-ferred heliographic longitudes on time scales of days, months or even longer wererecognized long.

The longitudinal distribution of sunspots and flares was initially studied in so-lar disc [Chidambara Aiyar, 1932; Evans et al., 1951; Behr and Siedentopf, 1952;

59

Fig. 5.1. Longitudes of two-day sunspots observed at first and last appearances [Chi-

dambara Aiyar, 1932] in solar disc reference frame. The thick (thin) solid line denotes

the first (last) appearance of Greenwich data and the dotted line denotes the first ap-

pearance of Kodaikanal data.

60

Fig. 5.2. Longitudinal distribution of total daily sunspot numbers summed over each

solar rotation in 27 13

day frame in 1903-1937 [Losh, 1939]. The horizontal axis denotes

the day of synodic solar rotation and the vertical axis is Wolf number.

61

Fig. 5.3. Longitudinal distribution of total daily sunspot numbers of each solar rotation,

using the same data as Losh [1939], in 27.2753 day frame for each solar cycle in 1903-1937

[Warwick, 1965].

62

Fig. 5.4. Active region chart [Dodson and Hedeman, 1975] shown by longitudes of large

plages with area of at least 1000 µHem in Carrington frame in 1970-1974. Time runs

from left to right. Vertical axis denotes Carrington rotation number, from top to bottom.

The ‘antecedents’ (‘descendents’) of each region are denoted by leftwards (rightwards)

and downwards shifted connecting lines. The size of the circle reflects the size of the

plage. Heavy dark circles denote flare-rich plages. A dark square outline represents a

region associated with proton events. Data are repeated for half a rotation on the right.

63

Dodson and Hedeman, 1964], and later in some rigidly rotating coordinate refer-ence frames, e.g., in Carrington frame [Warwick, 1965; Vitinskij, 1969; Trotter andBillings, 1962], in Bartels frame (with synodic rotation period of exactly 27 days)[Bumba and Howard, 1969; Balthasar and Schussler, 1983; Bai, 1990], or in otherrotation frame (e.g., 27 1

3 days in Losh [1939]). Accordingly, active longitudes werefound to appear at different longitudinal positions in different reference frames.Figure 5.1 shows the longitudinal distribution of first (thick-line) and last (thin-line) appearances of two-day spots in solar disc in 1874-1930 [Chidambara Aiyar,1932]. The frequency of their occurrence suggests a minimum between longitudes30 and 50 on either side of the Sun’s central meridian. The maximum frequencyof sunspot occurrence is found to be at different longitudes in reference frameswith different rotation period. Figures 5.2 and 5.3 show the longitudinal distri-bution of sunspot (Wolf) number in 1903-1937 in 271

3 -rotation-day frame [Losh,1939] and in 27.2753-day Carrington frame [Warwick, 1965], respectively. Theactive longitudes of sunspot activity were found more consistent in 27.2753-day(Carrington) frame than in 27 1

3 -rotation-day frame. Warwick [1965] also noticedthat proton flares leading to polar-cap absorption occurred mainly in two activelongitudes separated by around 180.

However, Wilcox and Schatten [1967] pointed out that the Carrington rotationperiod is not the rotation period of the Sun, and therefore one should take therotation period as a free parameter of the analysis, which needs to be determined byobservations. They reanalyzed the longitude of proton flare data used by Warwick[1965] and found that the active longitudes are more evident by using 28.0, 28.8,30.8, 31.9, and 32.9 days as the rotation period than using Carrington period.Following this idea by Wilcox and Schatten [1967], a number of other periodicitieswere found as well, such as, 12, 13.6, 31.63, 51, 73, 77, 103, 129, and 154 days[Knight et al., 1979; Bogart, 1982; Fung et al., 1971; Rieger et al., 1984; Bai, 2003b;Bogart and Bai, 1985; Bai and Cliver, 1990; Bai and Sturrock, 1991; 1993; Bai,1987b; 1992a;b; Ozguc and Atac, 1994].

So-called evolutionary charts of active regions and active-region family treesare also popular methods to study the longitude of long-lived and complex largeactive regions, in which solar flares occur quite often. Dodson and Hedeman[1968] noticed a band of active longitudes, whose Carrington longitude valuesincreased in time, suggesting a corresponding rotation period of 27d or slightlyless. They also noticed that strong solar activity occurs in areas separated by 180

in longitude. Studying the evolutionary charts in 1970-1974, Dodson and Hedeman[1975] confirmed that active longitudes rotate faster than Carrington frame andfound two active longitude bands centered around 90 and 270 in 1970-1971, butonly one active longitude band centered around 180 in 1973-1974 (see Fig. 5.4).

Recently, long-lived active longitudes of sunspots with the non-axisymmetryabout 0.1 were suggested by Berdyugina and Usoskin [2003] and the migrationof active longitudes was found to be in agreement with solar differential rotation.Although this result was criticized by Pelt et al. [2005] who claimed that the activelongitude pattern found by Berdyugina and Usoskin [2003] is an artefact of theused method. In the subsequent work Usoskin et al. [2005] analyzed the effectsof theirs method by using randomly generated data as the longitudes of sunspots.

64

Fig. 5.5. Longitudinal distribution of first appearance of (area weighted) sunspots in the

northern hemisphere in 1878-1996. a) Sunspots in the Carrington frame. Distribution is

nearly uniform. b) Sunspots in the dynamic reference frame. Non-axisymmetry is Γ =

0.11. c) Only one dominant spot of each Carrington rotation is considered, whence Γ =

0.19. d) The same as panel c) but averaged over half year in each of the two determined

active longitude bands, whence Γ = 0.43. The solid line in each panel represents the best

fit of double Gaussian [Usoskin et al., 2005].

65

Fig. 5.6. Active longitudes of X-class flares [Zhang et al., 2007a]. The longitudinal

distribution of X-flares in the dynamic reference frame for the time period of 1975-2005.

The horizontal axis denotes the longitude and the vertical the number of flares. The solid

line depicts the best fit of double Gaussian with two peaks at 90 and 270.

66

A large number of statistical tests showed that the average non-axisymmetry ofrandomly generated sunspots is 0.02 and the probability to obtain the 0.1 non-axisymmetry is less than 10−6. This implies that the persistent active longitudeswere not an artifact of method. If we correct the observed Carrington longitudesof sunspots according to the expected migration, the active longitudes shouldbe consistent in the new frame. Taking this concept into account Usoskin etal. [2005] found that two persistent active longitudes of sunspots separated by180 have existed during 11 cycles (1878-1996) with the level of non-axisymmetryabout 0.1 in a dynamic reference frame (see Fig. 5.5). If only the position of onedominant spot for each Carrington rotation is considered the non-axisymmetryincreases to 0.19. This explains why the previously found active longitudes arenot stable in any rigidly rotating reference frame. However, a systematic drift ofactive longitudes for sunspot numbers was not found by Balthasar [2007], probablybecause the sunspot numbers do not carry good longitudinal information, beingan index covering the whole visible solar disc.

A similar method to Usoskin et al. [2005] was employed by Zhang et al. [2007a;b]in the analysis of the longitudinal distribution of solar X-ray flares. They foundthat two active longitudes of X-ray flares have existed during three solar cyclessince GOES started X-ray observations in 1975. They also noted that the levelof non-axisymmetry increases with the X-ray intensity. The non-axisymmetry forall three classes of X-ray flares was found to be significantly larger than that forsunspots. The non-axisymmetry for powerful X-ray flares can reach up to 0.55(0.49) in northern (southern) hemisphere, implying that more than 70% of strongX-ray flares occurred in the two preferred active longitudes. Based on the activelongitudes and their migration with time in Carrington frame Zhang et al. [2008]developed a simple prediction of solar active longitudes.

However, the differential rotation parameters obtained for different solar tracerswere found to be quite different; even the parameters obtained for one class ofX-ray flares were different from those obtained for another class [Zhang et al.,2007b]. Note that in earlier studies [Usoskin et al., 2005; Zhang et al., 2007b], thetime resolution for the calculation of the shift of ALs was one Carrington rotationperiod. The migration M of ALs with respect to the Carrington reference framewas given by

M = Tc

Ni∑j=N0

(Ωj − Ωc), (5.1)

where N0 and Ni denote the Carrington rotation numbers of the first and the (i)throtation of the data set, Tc = 27.2753 days is the synodic Carrington rotationperiod, Ωj is the rotation rate of ALs, and Ωc is the angular velocity of Carringtonframe (in sidereal frame 14.1844 deg/day and in synodic frame 13.199 deg/day).The same value of M was used as the migration of ALs from the beginning tothe end of the (i)th Carrington rotation. However, even small difference betweenthe values of Ωj and Ωc (e.g., 0.2 deg/day) can cause a difference of up to a fewdegrees (5.5 deg) between the beginning and the end of a Carrington rotation.Thus, taking the same value of M in the beginning and the end of a Carringtonrotation could significantly lower the accuracy of the best fit rotation parameters.

67

In Paper I, we change the time resolution of M from one Carrington rotationto less than one hour. The time of M is taken the observation time of a flareor sunspot group transformed to a fractional day k of the Carrington rotation.Accordingly, the improved version of M has the form

M = Tc

Ni−1∑j=N0

(Ωj − Ωc) + k(ΩNi− Ωc) , (5.2)

where Ni−1 denotes the (i-1)th Carrington rotation number. Using this improvedmethod, we reanalyzed in Paper I the GOES X-ray flare data. The differentialrotation parameters for the three classes of flares were found to agree with eachother and the non-axisymmetry for the longitudinal distribution of flares was sys-tematically improved as well, which gives strong support for the success of theimproved method and the extracted parameter values. The non-axisymmetry ofX-class flares in the last three solar cycles can reach 0.59, suggesting that 79.5%of powerful flares occur in the two active longitudes. We also noticed that thenorthern and southern hemispheres rotate at significantly different average angu-lar frequencies during last three solar cycles.

In Paper II, using the improved method we studied the long-term variation ofthe differential rotation parameters of active longitudes of sunspots in 1874-2009,separately in the northern and southern hemispheres. The obtained rotation pa-rameters in the two hemispheres suggest a long-term period of variation of roughlyone century. The difference between the average rotating rates of the northern andsouthern hemispheres is in agreement with earlier studies [Bai, 2003a; Hoeksemaand Scherrer, 1987; Brajsa et al., 1997; 2000], but does not support the result ofGigolashvili et al. [2005] who claimed that the northern hemisphere rotates fasterduring the even cycles (20 and 22) while the rotation of southern hemisphere dom-inates in odd ones (cycles 19 and 21). (See Sect. 5.4.2 and Table 5.2 for moredetailed discussion about results on hemispheric difference found in Papers II, IIIand IV.)

Active longitudes have also been found in stellar activity of starspots, such as,spots on FK Com-type stars [Jetsu et al., 1991; 1993; 1999; Korhonen et al., 2002],on very active young solar analogues [Berdyugina et al., 2002; Jarvinen et al.,2005a;b; Berdyugina and Jarvinen, 2005], and on RS CVn stars [Berdyugina andTuominen, 1998; Lanza et al., 1998; Rodono et al., 2000]. Two permanent activelongitudes 180 apart seem to be a conspicuous pattern of stellar activity but cancontinuously migrate in the orbital reference frame. The nonlinear migration ofactive longitudes is a typical behaviour for single stars, young solar type dwarfs,and FK Com-type giants, which suggests the presence of differential rotation andsimilar changes in mean spot latitudes as on the Sun.

68

Fig. 5.7. Migration and flip-flop of active longitudes of sunspots [Berdyugina and Usoskin,

2003]. Semi-annual phases of the two active longitudes of sunspots in solar cycles 12-22 in

the northern hemisphere are plotted. Filled circles stand for the phases of the dominant

active longitude during the corresponding half year, and open circles stand for the less

active longitude. Curves depict the phase migration lagging behind the Carrington frame.

Vertical lines denote solar cycle minima. For better visibility, the migration restarts from

zero every second cycle.

69

Fig. 5.8. Longitudinal distribution of X-class flares in 1975-1985 in the dynamic reference

frame mentioned in Sect. 5.1. Two active longitudes are centered around 90 and 270,

180 apart. Six gathering areas of flares are denoted by numbers [Zhang et al., 2007a].

5.2. Flip-flop phenomenon

Periodic switching of the dominant activity from one active longitude to the otheris the so-called flip-flop phenomenon, which was first observed on the single, rapidlyrotating giant FK Comae [Jetsu et al., 1991]. Doppler images show that the activeregions evolve in size and indicate possible cyclic variations. While one activelongitude reduces its activity level, the other increases. When the active longitudeshave roughly the same activity level, a switch of the dominant activity from onelongitude to the opposite one occurs [Berdyugina et al., 1998; 1999; Korhonen etal., 2009]. Long time series of photometric data imply that flip-flops are regularlyrepeated on FK Comae, RS CVn stars, and young solar analogues, indicating aflip-flop cycle [Jetsu, 1996; Jetsu et al., 1999; Jarvinen et al., 2005b; Berdyuginaand Jarvinen, 2005]. A list of stars exhibiting a flip-flop phenomenon was recentlyreconstructed [Berdyugina, 2005].

The solar active longitudes also exhibit a flip-flop cycle. An analysis of 120years of sunspot data [Berdyugina and Usoskin, 2003] suggests that on the Sunthe alternation of the dominant spot activity between the two opposite longitudesoccurs periodically, at about 1.8-1.9-yr period on average, indicating 3.8-yr and3.65-yr flip-flop cycles in the northern and southern hemispheres, respectively.Transforming the Carrington longitudes (0-360) into phases (0-1), they calcu-lated the average phase every half a year, separately for the northern and southernhemispheres. Accordingly, a phase lag diagram of the two active longitudes withrespect to Carrington reference frame was obtained (see Fig. 5.7. for the phase for

70

the northern hemisphere). The migration continues over all 11 cycle and the activelongitude delay for about 2.5 rotations per cycle. Filled circles denote the phasesof the active longitude, which is dominant during the corresponding 6 months, andopen circles denote the less active longitude. Curves depict the migration of thephase lag, resulting from the change of the mean latitude of sunspots and differen-tial rotation. The active longitudes separated by 0.5 (180) in phase are persistentin both hemispheres for at least 120 years. The migration continues throughoutthe entire period and results in a phase lag of about 28 rotations for 120 years.One longitude dominates the activity typically for 1-2 years. Then a switch ofdominance to the other active longitude occurs rather rapidly. On an average, sixswitches of the activity occur during the 11-yr sunspot cycle and, thus, results inthe 3.7-yr flip-flop cycle, about 1/3 of sunspot cycle.

Similar flip-flop period was also found in heliospheric magnetic field [Takalo andMursula, 2002; Mursula and Hiltula, 2004] and solar X-ray flares [Zhang et al.,2007a;b]. Removing the migration of active longitudes due to differential rotation,the recovered longitude distribution of X-class flares (see Fig. 5.8) suggests thatthere are two persistent active longitudes centered around 90 and 270. There aresix groups of flare occurrence in Fig. 5.8 arranged around the two active longitudes.They demonstrate the periodical alternation of the dominance of active longitudesin cycle 21. This implies that a flip-flop cycle of flares is 1/3 of sunspot cycleperiod as well. A similar 1:3 relation between flip-flop and main activity cycle wasfound also in other stars [Berdyugina and Jarvinen, 2005].

5.3. Dynamo models

We are interested in the recent studies of the dynamo models for explaining theactive longitudes and flip-flops observed on the Sun and solar-like stars ratherthan the conventional solar dynamo theory for explaining the sunspot cycle andthe butterfly diagram.

The typical astrophysical dynamo is modeled by an electrically conducting ro-tating sphere, whose internal structure and motions are symmetric with respectboth to the rotation axis and to the equatorial plane. A model of this kind ex-cites different dynamo modes that have different symmetries and stabilities in thekinematic case.

The excited magnetic fields can be decomposed into two parts that are sym-metric (BmS ) and antisymmetric (BmA ) about the equatorial plane and have thefollowing form

BmA,S = Re[Cm

A,Sexp(imφ)], (5.3)

where CmA and Cm

S denote the antisymmetric and symmetric complex vector fieldswith respect to the equatorial plane. Both are symmetric about the rotation axis.m is a non-negative integer denoting the Fourier component with respect to theazimuthal coordinate φ (longitude) [Brandenburg et al., 1989; Radler et al., 1990].The total energy E of the B-field can be expressed as

71

Fig. 5.9. The solar dipole (A0 and S1) and quadrupole (A1 and S0) dynamo modes (pic-

ture taken from Fluri and Berdyugina [2004]). The four modes depict the distribution

of the poloidal magnetic field on the solar surface. The gray scale indicates the relative

strength of the magnetic field with white and black depicting opposite polarities. Assum-

ing the rotation axis in vertical direction, the A0 and S0 modes depict two axisymmetric

modes while A1 and S1 depict non-axisymmetric modes about the axis.

72

E =1

2µ

∫B2 × dV, (5.4)

where the integral is taken over the fluid body and outer space. A measure P ofthe degree of symmetry or antisymmetry of a B-field about the equatorial planeand a measure M for the deviation of a B-field from the symmetry about the axisof rotation can be defined by

P =ES − EAES + EA

(5.5)

and

M = 1− E0/E, (5.6)

respectively, where ES and EA are the total symmetric and antisymmetric partsof E from Eqs. 5.3 and 5.4, E0 is the energy in the axisymmetric part (M = 0) ofthe field. The evolution of solutions of the solar dynamo can be followed in a (P ,M) diagram with the four ‘corner solutions’ of (P , M) = (1,0),(-1,0),(1,+1),(-1,1),which are associated with the nonlinear continuation of the linear S0, A0, S1, andA1 modes, respectively [Radler et al., 1990; Moss, 2005]. The surface magneticfields of the S0, A0, S1, and A1 modes are displayed in Fig. 5.9. In slowly rotatingstars, like the Sun, axisymmetric modes (A0 and S0) are preferably excited. Inmore rapidly rotating stars the non-axisymmetric modes (e.g., A1 and S1) getexcited [Moss et al., 1995; Moss, 2005].

Besides the symmetry of the modes, their oscillatory properties are important.The mean-field dynamo theory favours oscillating axisymmetric modes with a clearcyclic behaviour and sign changes as in the sunspot cycle, while non-axisymmetricmodes appear to be rather steady. For explaining the flip-flop phenomenon, wherewe see both active longitudes and oscillations, an oscillating axisymmetric dynamomode needs to co-exist with the non-axisymmetric mode.

The possibility of such a mechanism was first demonstrated by Moss [2004]who obtained a stable solution with an oscillating axisymmetric mode and asteady, mixed-polarity non-axisymmetric mode. Moss [2005] studied the non-axisymmetric magnetic field generation in rapidly rotating late-type stars andproposed that a combination of S0 and S1 can produce the flip-flop. The super-imposed magnetic field structure of S1 and S0 is presented in Fig. 5.10. In thismodel if the non-axisymmetric field S1 remains constant and the axisymmetriccomponent S0 changes its sign and oscillates regularly every half cycle, the twoactive longitudes will switch by 180 at the same frequency of the S0 field. Theflip-flops occurring at the frequency of the sign changes of the axisymmetric modehave been observed in some RS CVn stars. More frequent flip-flops, as observedin single stars and the Sun, suggest a more complex field configuration. Fluriand Berdyugina [2005] suggested that a combination of one axisymmetric dynamomode (A0) and two non-axisymmetric modes (S1 and Y22 with m = 1 and m = 2,respectively) might explain the two solar active longitudes and the flip-flops. Fluriand Berdyugina [2004] showed that flip-flops could also occur due to alternationof relative strength of two non-axisymmetric (S1 and A1) modes, even without

73

Fig. 5.10. Superimposed magnetic field structure of S0 and S1. Solid lines depict the

S0 mode (quardrupole, axisymmetric) and dashed lines depict the S1 mode (dipole, non-

axisymmetric ) [Moss, 2005]. With the fields directed as indicated, the latitudinally

averaged field has a maximum at longitude 0 (on the left). If the direction of the S0

field is reversed, the maximum will appear at longitude 180 (on the right).

74

the sign change of any of the modes involved. Jiang and Wang [2007] studied thecondition for the excitation of the dominant axisymmetric mode and the conditionthat favours the non-axisymmetric mode and developed a coexistence of A0 modeand S1 mode which results in the flip-flop phenomenon. They pointed out thatthe axisymmetric magnetic field is mainly concentrated near the high latitude ofapproximately 55 at about 0.7R, where the radial shear of differential rotationis strong, while the non-axisymmetric field occur near the intermediate latitude of35 at the bottom of the tachocline, where the differential rotation is weak andthe diffusivity is low.

The solar dynamo could in principle generate non-axisymmetric magnetic con-figurations. However, a non-axisymmetric kinematic dynamo mode, even if prefer-ably excited, must rotate rigidly without exhibiting any effect of phase mix-ing, which results from the effect of the high differential rotation on the non-axisymmetric field. For a high differential rotation star as the Sun, the relativerotation between two nearby latitude belts stretches the non-axisymmetric fieldand leads to a differential rotation of the non-axisymmetric configuration. Thisis known as phase mixing [Bassom et al., 2005; Berdyugina et al., 2006; Usoskinet al., 2007b]. Phase mixing enhances the losses caused by turbulent diffusion.Therefore, the non-axisymmetric magnetic structures excited by various dynamosare expected to rotate rigidly.

A seeming interpretation for the rigidly rotating non-axisymmetric magneticfield and the differential rotation of active longitudes observed from sunspots wasproposed by Berdyugina et al. [2006] (described in Usoskin et al. [2007b]) in termsof the so-called the stroboscopic effect. The stroboscopic effect means that thenon-axisymmetric magnetic structure underlying the active longitudes is stronglyaffected by differential rotation, while the rotation law for the magnetic structureis still quasi-solid body. It gives the impression of a differential rotation of activelongitudes.

5.4. North-South asymmetry

Comparisons of activity in the two solar hemispheres show significant differences,which is called the north-south (N-S) asymmetry. The hemisphere with a highersolar activity is also called the active or preferred hemisphere [Hathaway, 2010].Besides the N-S asymmetry in solar activity level, N-S asymmetry is also found toexist in solar rotation. The variations in solar rotation are suggested to be mostlikely related to variations in solar activity [Javaraiah, 2003b; Javaraiah et al.,2005]. Recently, Javaraiah [2010] proposed that the mean meridional motion ofthe sunspot groups exhibits north-south difference as well. We give a brief reviewon the N-S asymmetry of solar activity and solar surface differential rotation inSect. 5.4.1 and in Sect. 5.4.2, respectively.

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5.4.1. N-S asymmetry in solar activity

As early as in 1890s Sporer [1889; 1890] and Maunder [1904] noted that therewere often long periods of time when most sunspots were found preferentially inone hemisphere. Significant N-S asymmetry was found in a large amount of solaractivity indices, such as the sunspot number and areas [Newton and Milsom, 1955;Waldmeier, 1971; Roy, 1977; Vizoso and Ballester, 1990; Carbonell et al., 1993; Liet al., 2000; 2002; Temmer et al., 2002; Vernova et al., 2002; Krivova and Solanki,2002; Knaack et al., 2004; Temmer et al., 2006b], the number of flares [Garcia,1990; Verma, 1987; 1992; Li et al., 1998; Temmer et al., 2001; Joshi and Pant, 2005;Joshi et al., 2006; 2007], the flare index [Atac and Ozguc, 1996; Joshi and Joshi,2004], prominences/filaments [Hansen and Hansen, 1975; Vizoso and Ballester,1987; Joshi, 1995; Verma, 2000; Gigolashvili et al., 2005], photospheric magneticfields [Howard, 1974; Rabin et al., 1991; Song et al., 2005], coronal EUV brightpoints [Brajsa et al., 2005], coronal green line intensity [Ozguc and Ucer, 1987;Storini and Sykora, 1995; Sykora and Rybak, 2005], and the heliospheric magneticfields [Mursula and Hiltula, 2003; 2004; Virtanen and Mursula, 2010]. Thus, thenorthern and southern hemispheres should always be considered separately whenstudying solar activity. Some results on the preferred hemisphere in cycles 8-23are displayed in Table 5.1.

However, quantifying the asymmetry itself is problematic. Taking the absoluteasymmetry index produces a strong signal around the solar maximum times, whiletaking the relative asymmetry index produces a strong signal around the minimumtimes. The absolute asymmetry ∆ and relative (or normalized) asymmetry δ canbe expressed as follows

∆ = An −As, (5.7)

δ =An −AsAn +As

, (5.8)

where An and As denote the activity index in northern and southern hemispheres,respectively. The high relative asymmetry found during solar minima is due toonly a few sunspot groups appearing on the Sun. If all of them are in the samehemisphere, the momentary absolute value of the relative asymmetry approachesto 1 [Swinson et al., 1986; Vizoso and Ballester, 1990].

Table 5.1 shows that during cycle 10, the N-S asymmetry shifts from northerndominance to the southern dominance and remains there until cycle 13. The N-Sasymmetry shifts back to the northern dominance in cycle 14 and continues to favorthe northern hemisphere until cycle 20. Southern dominance is restored in cycle 21and persist until recently. This implies that the N-S asymmetry of solar activityhas a long-term characteristic time scale of about 110 yr [Verma, 1993]. Long-termtrend in the N-S asymmetry has also been proposed [Newton and Milsom, 1955;Waldmeier, 1957; Bell, 1962; Vizoso and Ballester, 1990; Carbonell et al., 1993;Duchlev, 2001].

Besides the long-term periodicity of N-S asymmetry of solar activity, variationsat other periods of about 1 to 4 cycles [Vizoso and Ballester, 1987; 1989; Ozguc and

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authors tracer solar more activecycle hemisphere

Bell [1962] sunspot groups 8-9 North... sunspot groups 10-13 South

Li et al. [2002] sunspot groups 12-13 South... sunspot areas 12-13 South... sunspot numbers 12-13 South

Bell [1962] sunspot groups 14-18 NorthTemmer et al. [2006b] sunspot numbers 18-20 North

Li et al. [2002] sunspot groups 19-20 North... sunspot numbers 19-20 North... sunspot areas 19-20 North... flare index 19-20 North... sudden disappearances 19-20 North... of prominences ... ...

Verma [1987] major flares 19-20 North... radio bursts 19-20 North... white light flares 21 South... radio bursts 21 South... HXR bursts, CMEs 21 South

Temmer et al. [2006b] sunspot numbers 21-23 SouthGoel and Choudhuri [2009] polar faculae numbers 21-23 South

Table 5.1. Preferred hemispheres in cycles 8-23.

77

Ucer, 1987; Verma, 1993; Carbonell et al., 1993; Duchlev and Dermendjiev, 1996;Knaack et al., 2004; Javaraiah, 2008] and short-term periods of 1 to 5 yr [Ozgucand Ucer, 1987; Vizoso and Ballester, 1989; 1990; Verma, 1993; Knaack et al., 2004]were also found. Swinson et al. [1986] and White and Trotter [1977] claimed thatthere is a 22-yr period in the N-S asymmetry of solar activity. However, Carbonellet al. [1993] argued that the 22-yr period in the sunspot N-S asymmetry is notstatistically significant. The N-S asymmetry of solar activity within a solar cycle isrelated to the phase difference between the northern and southern hemispheres, i.e.,solar activity does not synchronously reach its maximum in the two hemispheres[Waldmeier, 1971; Temmer et al., 2006b; Li, 2008; 2009b; Li et al., 2010].

Carbonell et al. [2007] studied the N-S asymmetry in daily hemispheric sunspotnumber, X-ray flares, solar flare index, the yearly hemispheric number of activeprominences, the monthly hemispheric sunspot area, the hemispheric averagedphotospheric magnetic flux density and the total hemispheric magnetic flux ineach Carrington rotation. They found that many effects influence the statisticalsignificance of the N-S asymmetry, e.g., the type of data and data binning. Theyproposed different statistical tests for integer, dimensionless data (e.g., the dailyinternational sunspot numbers, or the daily X-ray flare numbers), non-integer,dimensionless data (e.g., the hemispheric sunspot numbers), and non-integer anddimensional data (e.g., flare index, magnetic flux in density). Appropriate databinning should be considered for different types of activity parameters.

It is believed [Goel and Choudhuri, 2009] that the Babcock-Leighton processof poloidal field generation is the source of randomness in the solar dynamo andthat the randomness of the Babcock-Leighton process may give rise to a strongerpoloidal field in one hemisphere compared to the other. Feeding the actual valueof the observed polar field at solar minima to a theoretical model based on meanfield equations, Goel and Choudhuri [2009] developed a numerical dynamo modelto explain the N-S asymmetry. They suggested that the asymmetry of the poloidalfield produced at the end of a sunspot cycle is the major factor determining theasymmetry of the next cycle.

5.4.2. N-S asymmetry in solar rotation

The N-S asymmetry in solar differential rotation has been reported by many au-thors [Gilman and Howard, 1984; Howard et al., 1984; Balthasar et al., 1986;Hoeksema and Scherrer, 1987; Antonucci et al., 1990; Japaridze and Gigolashvili,1992; Brajsa et al., 1997; 2000; Georgieva and Kirov, 2003; Georgieva et al., 2003;2005; Mursula and Hiltula, 2004; Javaraiah, 2003a; Nesme-Ribes et al., 1997; Gigo-lashvili et al., 2003; 2005; 2007; Wohl et al., 2010].

The measured rotation velocities differ from each other not only for differenttypes of solar activity formations, but they differ even for the same types as ob-tained by different authors. The discrepancy of the obtained velocities for differenttracers of solar activity can be explained by their different heights in the solar at-mosphere, where the radial differential rotation causes differences. Discrepancies

78

authors tracer solar fastercycle hemisphere

Javaraiah [2003a] sunspot groups 12 South... sunspot groups 13 North... sunspot groups 14-20 South

Gigolashvili et al. [2003] Hα filaments 20, 22 North... Hα filaments 19, 21 South

Scherrer et al. [1987] PMF 20-21 NorthHoeksema and Scherrer [1987] coronal magnetic field 21 North

Antonucci et al. [1990] PMF 21 NorthBrajsa et al. [1997] Hα filaments 21-22 North

... LBT regions 21-22 NorthBrajsa et al. [2000] LBT regions 21-22 NorthKnaack et al. [2005] PMF 21, 23 North

... PMF 22 SouthBai [2003a] major flares 19-23 North

Wohl et al. [2010] SBCS 23 SouthPaper II sunspot groups 12 North

... sunspot groups 13-16 South

... sunspot groups 17-23 NorthPaper III sunspot groups 13-16 South

... sunspot groups 17-22 North

Table 5.2. Faster rotating hemisphere in cycles 12-23. LBT denotes low-brightness-temperature, PMF photospheric magnetic fields, and SBCS small bright magneticstructures.

among the same types of tracers may be due to different time binning of data.Increasing evidence [Balthasar et al., 1986; Bai, 2003a; Knaack and Stenflo, 2005;Knaack et al., 2005; Balthasar, 2007; Heristchi and Mouradian, 2009] suggests thatthe solar differential rotation changes from cycle to cycle and even within one cycle.Bouwer [1992] claimed that precise periods between 27 and 28 days persist onlyfor a short time, sometimes only for a few solar rotations. Howard and Harvey[1970] found that the differential rotation even varies day by day. Therefore, boththe depth of the phenomenon in the solar atmosphere and the time of occurrenceshould always be taken into account when studying solar differential rotation andits variation.

Balthasar et al. [1986] and Balthasar [2007] studied the long-term variation ofsolar differential rotation using sunspot groups and sunspot numbers as tracers.However, they did not separate northern and southern hemispheres. The long-termvariation of differential rotation from cycle to cycle, separated for the northern andsouthern hemispheres, has not been widely studied yet. The obtained results sofar are listed in Table 5.2. The results obtained by different authors significantlycontradict each other.

79

In Paper II, we studied the long-term variation of the solar surface rotationusing the improved analysis mentioned in Sect. 5.1 and sunspot observation in1874-2009. This paper confirms the result [Bai, 2003a; Hoeksema and Scherrer,1987; Brajsa et al., 1997; 2000] that the northern hemisphere rotates faster thanthe south in the last five solar cycles and suggests that the long-term period ofN-S asymmetry of solar rotation is around 11 cycles.

Heristchi and Mouradian [2009] and Plyusnina [2010] analyzed the variation ofsolar differential rotation in northern and southern hemispheres with time, butfound no N-S asymmetry in cycles 19-23. This is probably due to the tracers theyused. Heristchi and Mouradian [2009] used the daily sunspot numbers, which donot carry sufficient longitude information since they are smoothed over the wholesolar disc. Similarly, Plyusnina [2010] used the daily values of sunspot groupareas summed over the whole disc and also over each hemisphere. The data wassmoothed in large longitude windows from 30 to 90.

Javaraiah [2003a] found that the variation of A (equator rotation angular ve-locity, see Eq. 3.2) has a 90-yr periodicity and B (latitude gradient) has a 100-yrperiodicity in the northern and southern hemispheres, respectively. Such long-term periodicity of the differential rotation parameters was also found in in thecombination of sunspots in the two hemispheres [Javaraiah et al., 2005]. This sug-gests a long period of N-S asymmetry of solar rotation in time scale of a hundredyears. The long-term periodicity of the variation of solar surface rotation is gener-ally thought to be related to the long-term trend of solar cycle (i.e., the Gleissbergcycle), which may result from the coupling of the Sun’s spin and orbital motionswith respect to the center of mass of the solar system.

Shorter periods (17-yr, 22 yr, 47-yr, etc.) of N-S asymmetry of solar rotationhave also been proposed [Javaraiah, 2003a;b; Georgieva and Kirov, 2003; Gigo-lashvili et al., 2005]. Gigolashvili et al. [2005] found that the northern hemisphererotates faster during even cycles (20 and 22) while the rotation of southern hemi-sphere dominates in odd ones (cycles 19 and 21). Changes in the northern andsouthern hemispheres could be interpreted as an oscillation with a 22-yr period.However, most other results listed in Table 5.2 do not support their observations.

In Paper II, we noticed that the best fit rotation parameters obtained for longerfit intervals, e.g., 3-cycle and 6-cycle intervals, have less uncertainties and moreconsistent variation than the parameters obtained for shorter interval such as 1-cycle interval (see Table 5 and Fig. 5 in Paper II). The rotation parametersobtained for sunspots in last three cycles in each hemisphere in Paper II agreewell with those obtained for X-ray flares for the same time period in Paper I. Thisgives further support to the improved method.

In Paper III, we reanalyzed the sunspot data in 1874-2009 using 3-cycle runningfit intervals. The results show that the average rotation rates in each hemispherevary quite similarly during the whole time period as when using 1-cycle intervals.In earlier studies, the weight of sunspot area was used in the merit function tosearch for the best fit rotation parameters, giving more weight to large spots.Accordingly, the rotation parameters obtained , e.g., in Paper II are closer to therotation of large sunspots. To study the effect of large spots, the weight in themerit function was removed in Paper III. We found that there was little difference

80

between the values of rotation parameters obtained in Paper II with weight and inPaper III without weight, and that the long-term variation of rotation in the twohemispheres agrees well with each other. The faster rotating hemispheres remainthe same in the two papers during the whole time period (see Table 5.2).

Fig. 5.11. The yearly rotation rates Ω17 at the average latitude of 17. in the northern

(filled circles) and southern (open circles) hemispheres obtained for 3yr-running fit inter-

vals and no weighting, and smoothed over 11 points (a solar cycle). The horizontal line

denotes the mean value of Ω17 over the whole time period.

In Paper II, we also found that the non-axisymmetry of sunspots decreaseswith increasing fit interval. This decrease most likely results from the long-termvariation in solar rotation. In Paper IV, we studied the long-term variation insolar rotation using short fit intervals such as 1-yr, 3-yr and 5-yr. The long-termvariation in rotation parameters obtained for 1-yr, 3-yr and 5-yr intervals also showthe same pattern presented in Fig. 5.11. Figure 5.11 shows the yearly rotationrate at the average latitude of sunspots of 17 in each hemisphere obtained using3-yr running fit intervals. The rotation parameters obtained for each 3-yr periodare used for the middle year. The yearly parameters are smoothed over a solarcycle to show the variation pattern more clearly. The long-term evolution of therotation in the two hemispheres obtained using such short fit intervals also show ananti-correlation. Comparing Fig. 5.11 with the right-hand side panels of Figure5 in Paper II, one can notice that the only disagreement between the rotationparameters obtained for 3-yr intervals and those obtained for 3-cycle intervalsexists in the first 3-cycle period (1890-1910). This is probably related to theextremely weak and long solar minima, during which the short interval fits mayeasily lose the continuous evolution of phase. During cycles 15-22, the rotationof the northern hemisphere keeps increasing while the southern hemisphere keepsdecreasing. The results for this period agrees well between short and long fitintervals.

81

Fig. 5.12. North-south asymmetry (N-S)/(N+S) in Ω17 obtained from Fig. 5.11.

The difference between the rotation rates of northern and southern hemispheresobtained for 3-yr fit intervals is shown in Fig. 5.12. The two peaks of the differencein the rotation of the two hemispheres at about 1900 and 1990 suggest a long-termperiodicity in N-S asymmetry of about 90 years. This supports the results ofJavaraiah [2003a] who also found a 90-yr periodicity in the differential rotationparameters.

6. Summary

In this thesis we studied solar active longitudes of X-ray flares and sunspots. Wenoted that in earlier work the migration of active longitudes with respect to theCarrington rotation was based on rather coarse treatment. We improved the ac-curacy to less than one hour. Accordingly, not only the rotation parameters foreach class of solar flares and sunspots are found to agree well with each other,but also the non-axisymmetry of flares and sunspots is found to be systematicallyenhanced. The non-axisymmetry of X-class flares in the last three solar cyclescan reach 0.59, suggesting that 79.5% of powerful flares occur in the two activelongitudes. This hightlights the importance of prediction of active longitudes ofsolar flares.

Using the improved analysis, the spatial distribution of sunspots in 1876-2008 isstudied. The non-axisymmetry of sunspots is found to be slightly improved. Thestatistical evidence for different rotation in the northern and southern hemispheresis greatly improved by the revised treatment. Moreover, we have given consistentevidence for the periodicity of about one century in the north-south difference.We found that the active longitudes of sunspots are well determined by short timeintervals due to the variation of solar rotation. The non-axisymmetry of activelongitudes determined by 3-yr fit intervals is significantly larger than obtained by11-cycle interval.

In the future work, we will study the correlation between the active longitudes ofsunspots and the active longitudes of flares at short time scales, and we reexaminethe flip-flop phenomenon using the improved analysis.

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