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 fact
that solar activity does not occur uniformly at all heliographic longitudes was noticed
by Carrington as early as in 1843. The longitude ranges where solar activity occurs
preferentially are called active longitudes. Active longitudes have been found in various
manifestations of solar activity, such as sunspots, flares, radio emission bursts, surface
and heliospheric magnetic fields, and coronal emissions. However, the active longitudes
found when using different rigidly rotating reference frames differ significantly from each
other. One reason is that the whole Sun does not rotate rigidly but differentially at
different layers and different latitudes. The other reason is that the rotation of the Sun
also varies with time.
Earlier studies used a dynamic rotation frame for the differential rotation of the Sun
and found two persistent active longitudes of sunspots in 1878-1996. However, the migration of active longitudes with respect to the Carrington rotation was treated there rather
coarsely. 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 found
to agree well with each other, but also the non-axisymmetry of flares and sunspots is
systematically increased.
We also studied the long-term variation of solar surface rotation. Using the improved
analysis, the spatial distribution of sunspots in 1876-2008 is analyzed. The statistical
evidence for different rotation in the northern and southern hemispheres is greatly improved by the revised treatment. Moreover, we have given consistent evidence for the
periodicity 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 my
adoptive parents for all their love and support! They have
been and will always be great sources of encouragement
and inspiration throughout my life!
3
Acknowledgements
This work has been carried out in the Department of Physics at the University
of Oulu, Finland. First, I would like to express my deepest gratitude to Prof.
Ilya Usoskin. He very enthusiastically answered my questions at our first meet
during the Beijing COSPAR 2006. He has been supporting me with my study
and 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 the
guidance of Prof. Kalevi Mursula and himself. In the beginning of my first visit
to Oulu, Ilya offered me plenty of help and guidance for managing the new daily
life, 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 been
giving me plenty of support and encouragement to deal with the problems I have
had during doing the thesis. Accordingly, the scientific significance of the thesis
has been greatly improved. I also sincerely appreciate his fatherly advices on my
life.
I would like to express my deepest gratitude to my Beijing chief Prof. Huaning
Wang, a great scientist in solar physics and solar activity prediction. He has been
guiding me through my science career since the very beginning and giving me
continuous support. He always reminds me to find out what is the physics behind
the phenomenon that I see. The other members of my Beijing group are also
deeply acknowledged, especially Prof. Zhanle Du, Xin Huang, Jian Gao, Yanmei
Cui 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 University of Oulu, the Academy of Finland, the European Commission’s Seventh
Framework Programme for eHeroes project, and the Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences.
I fully acknowledge the pre-examiners of my thesis Prof. Roman Brajša and
Assoc. Prof. Dimitri Ponyavin for their valuable comments and advice. I would
5
also like to express my sincere appreciation to Dr. Rainer Arlt who agreed to spare
his time to be the opponent.
I wish to express my warmest gratitude to all the other members in the Space
Physics group for their friendship and help, Reijo Rasinkangas, Leena Kalliopuska,
Olesya Yakovchouk, Alessandra Pacini, three Virtanen brothers, Timo Asikainen,
Petri Kalliomäki, Tomi Karppinen, Lauri Holappa, Ville Maliniemi, Anita Aikio,
Ritva Kuula, Tuomo Nygrén, Kari Kalla, Timo Pitkänen, Lei Cai, Perttu Kantola
and Raisa Leussu. Members of the Geophysics group are also acknowledged for
their warm companionship during the coffee break.
In the end I would like to thank my dear sister and brother for their constant
support and love during my study and my life. I definitely must thank my beloved
husband Ari Ronkainen for everything. He has been strongly supporting me for
my work and for my hobbies. I would also like to thank my husband’s parents and
relatives 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 of
active longitudes of solar X-ray flares, J. Atmos. Sol. Terr. Phys., 73, 258263, 2011.
II
L. Zhang, K. Mursula, I. G. Usoskin, and H. N. Wang, Global analysis of
active longitudes of sunspots, Astron. Astrophys., 529, A23, 2011.
III L. Zhang, K. Mursula, I. Usoskin, H. Wang, and Z. Du, Long-term variation
of 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 the
hemispheric asymmetry of solar rotation, Astron. Astrophys., submitted.
In the text, original papers are referred to using roman numerals I-IV.
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Original publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . .
2. Solar activity . . . . . . . . . . . . . . . . . . .
2.1. Sunspots . . . . . . . . . . . . . . . . . . .
2.1.1. Wolf sunspot numbers . . . . . . .
2.1.2. Group sunspot numbers . . . . . .
2.1.3. Sunspot areas and positions . . . .
2.1.4. Cyclicities in solar activity . . . .
2.1.5. Grand minima and maxima . . . .
2.2. Solar flares . . . . . . . . . . . . . . . . .
2.2.1. Classification of solar flares . . . .
2.2.2. Phases of a typical flare . . . . . .
2.2.3. Radio emissions during solar flares
2.2.4. Flare models . . . . . . . . . . . .
2.3. Coronal mass ejections . . . . . . . . . . .
2.4. Solar energetic particles acceleration . . .
3. Solar rotation . . . . . . . . . . . . . . . . . . .
3.1. Heliographic coordinates . . . . . . . . . .
3.2. Surface rotation . . . . . . . . . . . . . . .
3.3. Internal rotation . . . . . . . . . . . . . .
3.4. Torsional oscillation . . . . . . . . . . . .
3.5. Meridional circulation . . . . . . . . . . .
4. Active latitudes . . . . . . . . . . . . . . . . . .
5. Active longitudes . . . . . . . . . . . . . . . . .
5.1. History of active longitude research . . . .
5.2. Flip-flop phenomenon . . . . . . . . . . .
5.3. Dynamo models . . . . . . . . . . . . . . .
5.4. North-South asymmetry . . . . . . . . . .
5.4.1. N-S asymmetry in solar activity .
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5.4.2. N-S asymmetry in solar rotation . . . . . . . . . . . . . . .
6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
82
1. Introduction
As is well known, the Sun is the source of energy that supports the existence of life
on 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 is
far more active when observed through telescopes, especially through the modern
space-borne instruments with high temporal and spatial resolution. Even the quiet
Sun appears very active in the observations of the newest space solar telescope Solar Dynamics Observatory (SDO). Small scale tornadoes or cyclones are found
to 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 to
be associated with microflares. A huge tornado can be observed before a coronal
mass ejection (CME) (see the right panel of Fig. 1.1).
There are gradually changing activities and bursting activities on the Sun. The
former refer to activities which vary gradually in size and energy and last a relatively long time. These include sunspots, plages, faculae, quiet filaments and
prominences, coronal holes and streamers, solar wind, and the general radiation.
The latter refer to explosive events with sudden and huge energy release. These
events include, e.g., flares, CMEs, and explosive filaments and prominences.
Solar activity affects the near-Earth space in many different ways. Variation in
solar conditions at short time scales, e.g., from a few seconds to days is involved
with space weather, while the long-term conditions, from several months to one
solar cycle and longer, relate to space climate. Space weather effects are mainly
caused by explosive transient solar events, i.e., flares and CMEs. The various
types 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. Solar
energetic particles (SEPs), which are accelerated up to relativistic velocities in a
flare or by CME shock waves, can cause a disruption of technical systems and
may threaten the life and health of astronauts working in the space outside the
protection of the Earth’s magnetosphere. Gigantic clouds of ionized gas ejected
in a CME can cause a series of geomagnetic effects when they reach and hit the
Earth’s magnetosphere. These effects can cause malfunctions in satellite systems.
They cause the ionosphere to more effectively absorb high frequency radio waves,
12
Y (arcsec)
-580
-600
-280
-260
X (arcsec)
Fig. 1.1. Tornadoes on the Sun observed with SDO/AIA 171Å 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 solar
induced technological damage was noticed in the telegraph transmission due to a
white flare observed by Carrington and Hodgson in 1859. This event was widely
spread through newspapers and articles, which caught much public attention and
became 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 become
more and more serious. In order to lessen this damage, sciences and techniques
involved in these processes have to be improved. On one hand, we have to seek out
signatures of indication of a powerful solar event and the connections of solar events
and the space and near-Earth environment so that we can identify the geoeffective
events from those which are not geoeffective. This study has developed into space
weather and forecasting sciences. Although the Solar-terrestrial physics started
long ago, a large amount of connections and physical mechanisms involved with
solar activity and space effects are still open. The signatures of a geoeffective event
have not been surely identified yet. On the other hand, we need to estimate how
serious the effects from a solar event can be so that the corresponding protective
measures 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-terrestrial
environment has been roughly established. Space weather, nowadays, deals with
activity conditions on the Sun and their effects on the near-Earth space, especially
13
what might be hazardous to the technological systems and endanger humans’ life
and health both in the spacecraft and on the ground.
At the present time the parameters of solar wind (the material in the interplanetary space) can only be observed at the first Lagrangian point (L1 ∼1.5·106 km
away to the Earth, 1/100 of the Sun-Earth distance). This leaves us only 30 to
60 minutes before the plasma reaches the Earth. ACE (Advanced Composition
Explorer launched in 1997), SOHO (Solar and Heliospheric Observatory launched
in 1995) and WIND (Global Geospace Science WIND moved to L1 in 2004) at the
L1 point measure real time solar wind parameters, such as the magnetic field, the
velocity density and temperature of electrons and protons.
International Space Environment Service (ISES) consisting of 13 Regional Waring Centers (RWCs) all over the world, and the European Space Agency (ESA)
collect space-borne and ground-based space-weather related observations and forecasts and provide the data to users. The Space Weather Prediction Center of
NOAA (National Oceanic and Atmospheric Administration of USA) performs 3day 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 leads
that only those charged particles accelerated in the flares occurring close to the
western limb of the solar disc may hit the Earth. Magnetic clouds from CMEs with
inclosed magnetic structure propagate radially from the Sun to the interplanetary
space. Thus, CMEs may reach the near-Earth space when they appear close to
the center of solar disc. Those events which occur on the back side of the Sun are
not geo-effective. Therefore, the location of eruptions on the Sun is an important
factor to space weather.
It was noted long ago that solar activity does not occur uniformly at all longitudes on the Sun. Those longitude ranges where solar activity appear more often
than elsewhere are called active longitudes. Active longitudes (ALs) were found
in various manifestations of solar activity, such as sunspots, flares, surface magnetic fields. However, the suggested ALs differ with each other significantly in
rigidly rotating reference frame. This is because the Sun does not rotate rigidly
but differentially with latitude and depth, showing the so-called surface and radial
differential rotation. Accordingly, different types of solar activity appearing at
different height in the solar atmosphere, e.g., sunspots at solar surface and CMEs
in the high corona, rotate at different rates. And the rotation of one type of solar
activity may change due to its latitudinal variation in a solar cycle.
Taking solar surface differential rotation into account Usoskin et al. [2005] introduced a dynamic differentially rotating reference frame. In this frame, the
shift of ALs due to the surface differential rotation was removed. Two persistent active longitudes of sunspots were found to last for 11 cycles. Using similar
reference system two ALs of X-ray flares were found to exist for 3 cycles. The
non-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% of
powerful flares occur in the two active longitudes. This implies the importance of
prediction of active longitudes of solar flares. However, the rotation parameters
obtained for each class of solar flares differ with each other significantly.
14
In order to find out the source of the discrepancy among the rotation parameters
for sunspots and each class of flares, the locations of sunspots and X-ray flares in
last three solar cycles were reanalyzed in this thesis.
Rotation rate of the Sun varies also with time. Increasing evidence [Howard and
Harvey, 1970; Knaack et al., 2005; Balthasar, 2007] suggests that the rotation rate
of solar surface changes from cycle to cycle, even within one cycle. The rotation
rate in the northern hemisphere was found to vary differently with the southern
hemisphere showing N-S asymmetry. The suggested faster rotating hemispheres
in each solar cycle by different authors disagree with each other. The locations of
sunspots in 1876-2008 are analyzed in this thesis in order to study the long-term
variation 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 are
known 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 solar
disc observed on Dec 11, 2006 by SOHO (see Fig. 2.1 ). Although the details of
sunspot generation are still a matter of research, sunspots are generally thought to
be generated by toroidal magnetic fields lying below the photosphere and brought
to the surface by buoyancy [Kivelson and Russell, 1995; Rigozo et al., 2001].
Registered sunspot observations have been dated at least back to the fourth
century BC. Irregular naked-eye observations of sunspot activity can be found in
Chinese 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
[Hsü, 1972; Price et al., 1992]. It was shown that the oriental historic data really
reflect 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 that
ancient astronomers often mixed real sunspots with other celestial or meteorological phenomena; Second, ancient sunspot observations were not systematic and, as
a result, are non-uniform in time; Third, some climatic effects may be present in
different ancient sunspot series [Willis et al., 1980].
Sunspots were first telescopically observed in 1610 by the English astronomer
Thomas Harriot and the German (Frisian) father and son, David and Johannes
Fabricius. David and Johannes Fabricius noted the rotation of sunspots and the
Sun and published a description on the rotation in 1611. The rotation of the Sun
and sunspots was independently discovered by Christoph Scheiner and Galileo
Galilei as well in 1610 by observing sunspots continuously for a few days. They
both published their observations in 1612. Using an improved telescope, Christoph
Scheiner measured the rotation rate of sunspots near the equator in 1630. Unfortunately, not many sunspots were recorded in the following hundred years since
16
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 during
1645-1715. Since the invention of telescope, daily sunspots have been observed
for nearly four hundred years. Sunspot number series (see later) is the most used
index 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 be
observed in detail. Most sunspots are observed to contain two parts known as the
umbra and the penumbra. The umbra is at the center and the darkest part of a
sunspot and has the strongest and vertical magnetic field of the sunspot, while the
penumbra, surrounding the umbra, is a little lighter and its magnetic is weaker and
nearly horizontal. Figure 2.2 depicts the detected structure of the sunspot group
shown in Fig. 2.1 observed by Hinode Satellite. Accordingly, the areas of the
umbra and penumbra can be obtained. The continuous measurement of sunspot
areas began at Greenwich Observatory in 1874.
17
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 Zürich 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 by
Rudolf Wolf of Zürich Observatory in 1850s. Wolf started daily observations of
sunspots in 1848 soon after Heinrich Shwabe discovered the sunspot activity cycle
with 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 importance
of 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 in
all groups visible on the solar disc and k denotes the individual correction factor
which compensates differences in observational techniques and instruments used
by different observers, thus normalizing the different observations to each other
[Wilson, 1992].
Rudolf Wolf worked at Zürich Observatory as the “primary” observer (judged
as the most reliable observer during a given time) in 1848-1893. He extended the
series back to 1749 by using the observations from the primary and secondary
observers and filled in the gaps by using linear interpolation and geomagnetic declination measurements as proxies. Therefore, Wolf sunspot numbers are a mixture
of 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, Zürich),
18
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. Using the observations from a single observer aims to make Rz a homogeneous time
series.
The Rz index was provided and maintained by the Swiss Federal Observatory in
Zürich, Switzerland through 1980. Since 1981 to the present the Rz index has been
provided and maintained by the Sunspot Index Data Center in Brussels, Belgium
[Hathaway et al., 2002]. The data process was changed in 1981 from using the
observation of the primary observer to using a weighted average of observations
by many observers when the Royal Observatory of Belgium took over the process
of preparing the International sunspot numbers from the Zürich observatory.
Yearly values of Rz series are available since 1700, monthly values start from
1749 and daily values from 1818 [Friedli, 2005]. However, the data is reliable
only since Wolf’s time, i.e., 1848. The earlier data which was reconstructed by
Wolf has pretty large uncertainties due to serious data gaps and the different
calculation techniques used by observers. Data gaps of a few days in 1818-1848
are manageable. Data gaps of many months in 1749-1848 and of a few years in
1700-1749 were filled by Wolf. There are many filled values in 1700-1817 resulting
in a rather poor sunspot index during this period. Moreover, Wolf obtained the
daily values from his correspondents without checking the original observations
himself. Thus significant uncertainties exist in the definition of G and N due to
different observers. Vitinsky et al. [1986] estimated the systematic uncertainties to
be about 25% in monthly sunspot numbers. The quality of the data is considered as
‘hardly reliable’ before 1749, ‘questionable’ during 1749-1817, ‘good’ during 18181847, and ‘reliable’ from 1848 onwards [Eddy, 1976; McKinnon and Waldmeier,
1987; Kane, 1999]. The Rz index provided the longest continuous measure of solar
activity [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 remove
variations associated with the Sun’s 27-day synodic rotation period. The monthly
values still have large oscillations and require additional averaging processes to
produce a signal which varies smoothly from month to month. A commonly used
averaging method is the 13-month running mean of the monthly values. If Rn is
the monthly value for month n, then the 13-month running mean is given by
5
X
1
Rn =
(Rn−6 + 2
Rn+i + Rn+6 ),
24
i=−5
(2.2)
simply called the smoothed sunspot number [Hathaway et al., 1999]. Solar cycle
maxima and minima are usually determined by these smoothed monthly values.
The period between two successive minima defines the solar cycle. Solar cycle 1
started 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 cycle
maxima in order to suppress the noise when studying long-term variations of solar
19
activity [Gleissberg, 1944; Soon et al., 1996; Mursula and Ulich, 1998]. The Gleissberg filtered solar maximum of cycle n [Gleissberg, 1944; Petrovay, 2010] has the
form
1 n−2
n−1
n
n+1
n+2
(Rmax )nG = (Rmax
+ 2Rmax
+ 2Rmax
+ 2Rmax
+ Rmax
).
(2.3)
8
2.1.2. Group sunspot numbers
Noting that no error estimates are available in the Rz series and some early
sunspot observations were not included in Wolf’s collection, Hoyt and Schatten
[1998a;b] collected many more observations than Wolf did, and introduced the
Group sunspot numbers since the observations from a few early observers have only
the daily number of sunspot groups, but not the number of individual sunspots.
The daily Group sunspot number RG is defined as
RG =
12.08 X 0
ki Gi
N
i
(2.4)
where Gi is the number of sunspot groups recorded by the ith observer, ki0 is the
ith observer’s individual correction factor, N is the number of observers used to
form a daily value, and 12.08 is a normalization number scaling RG to Rz values
in 1847-1976. The normalization number varies slightly due to the total number
of observations used for different time period.
The main importance of the RG series is in extending the sunspot number observations 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 collect the observations before 1874, when the Royal Greenwich Observatory started
recording detailed sunspot observations including their position and area. The
database for RG has about 80% more daily observations than Rz for the period of
1610-1874. Therefore, the RG series is more reliable and homogeneous than the
Rz series, especially before 1874. The two series agree closely with each other since
1950s [Hoyt and Schatten, 1998b; Letfus, 1999]. The main characteristics of solar
cycles obtained from the two time series also agree with each other [Hathaway et
al., 2002].
The RG series includes not only one primary observation but all available observations. This approach allows to estimate systematic uncertainties of the resulting
RG . The systematic uncertainties for the yearly RG values are caused by four
aspects: missing observations; uncertainties in the values of k 0 ; random errors in
the daily values; and drifts in the k 0 values. The final systematic error is about
10% 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 to
RG in 1700-1882 is 25-50% lower than that given by Wolf. Consequently RG sug-
20
gests that solar activity in the last few decades has been higher than it was during
several earlier centuries.
The RG series provides a pure sunspot series without filling in missing months
or years. The number of days used to form the monthly mean values is also
available for the series. This allows users to evaluate the data coverage for each
period and to estimate related errors. The RG series has still some uncertainties
and inhomogeneities [Letfus, 2000]. However, the great advantage of this series
compared 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 interval there are 32 years when RG has an unreliable value (uncertainty greater than
25%) or is missing. The RG series has not been continued beyond 1995. There are
minor problems in the work of Hoyt and Schatten [1998b]. E.g., the Julian calendar dates were taken to be Gregorian in some cases, and individual sunspots were
occasionally defined as sunspot groups [Vaquero, 2007]. Also some published observations were not covered in this work. Recently a great amount of old drawings
of 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 the
Boulder sunspot numbers and the American sunspot numbers. These additional
numbers can be scaled to the International Sunspot Numbers by using appropriate
correction factors [Hathaway, 2010].
2.1.3. Sunspot areas and positions
Another widely used solar activity index is the sunspot areas. In May, 1874 the
Royal Greenwich Observatory (RGO) started compiling sunspot observations from
a small network of observatories including Greenwich, England; Cape Town, South
Africa; Kodaikanal, India; and Mauritius to produce daily sunspot positions and
areas. Sunspot areas were measured in units of millionths of a solar hemisphere
(µHem) by dividing the sunspot in photographs into small squares. Both umbral
areas and whole spot areas were calculated. The measured areas were corrected for
foreshortening on the visible disc. The accuracy of sunspot areas was determined
up to 0.1 µHem. Sunspot positions were measured relative to the center of solar
disc, i.e., latitude and central meridian distance. The accuracy of sunspot positions
was determined to 0.1 degrees. The RGO stopped preparing its data in 1976 when
the US Air Force (USAF) started compiling sunspot area and position data from
its Solar Optical Observing Network (SOON) in 1977 with the help of the US
National Oceanic and Atmospheric Administration (NOAA).
The sunspot areas reported by USAF/NOAA are derived from sunspot drawings
prepared at the SOON sites, which were initially measured by overlaying a grid
and 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 areas
are determined by rounding to the nearest 10 µHem. The combined dataset of
RGO and USAF/NOAA SOON, from May 1874 to the present, is available online
at http://solarscience.msfc.nasa.gov/greenwch.shtml. The average daily sunspot
area over each solar rotation is shown in the bottom panel of Fig. 4.1.
Wilson and Hathaway [2006] have studied the measurements taken in RGO
and SOON in detail. They also studied the relationship between sunspot area and
sunspot number among RGO, SOON, Rome Observatory, Swiss Federal Observatory and Royal Observatory of Belgium. The relationship between total corrected
sunspot area and sunspot number is not good for daily values, but becomes better when using monthly or yearly averages. However, the sunspot areas reported
by USAF/NOAA are significantly smaller than those given by RGO [Fligge and
Solanki, 1997; Baranyi et al., 2001; Hathaway et al., 2002]. Figure 2.3 obtained by
Hathaway [2010] shows the relationship between sunspot area and sunspot number
using 13-month running mean values. Top panel shows that the smoothed spot
area reported by RGO and the smoothed International sunspot number have very
high correlation (r = 0.994, r2 = 0.988). The best fit to the two quantities gives
a slope of 16.7 and an offset of about 4. The zero point for the sunspot number is
shifted slightly from zero because a single sunspot does not give a sunspot number
of 1, but 11 (see E.q. 2.1) for a correction factor k = 1.0. Therefore, the relationship between the RGO sunspot area ARGO and sunspot number Rz is roughly
described as
ARGO = 16.7Rz .
(2.5)
Bottom panel shows the relationship between the smoothed USAF/NOAA SOON
sunspot 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 earlier
RGO sunspot areas, the USAF/NOAA SOON sunspot areas should be multiplied
by 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 large
series of sunspot positions was obtained by Arlt [2008; 2009a] for the years 17491796 from the drawings of Johann Staudacher. A number of positions of sunspots
were determined by Arlt [2009b] from the observations at Armagh Observatory in
1795-1797.
2.1.4. Cyclicities in solar activity
The most prominent periodicity of sunspot activity is the 11-year cycle apart from
the Sun’s 27-day rotation period. The 11-year cycle is also called the Schwabe
cycle, discovered by Schwabe [1844] using a number of daily sunspot groups in
1826-1843. This cycle dominates the sunspot activity during almost the whole
22
Fig. 2.3. The relationship between sunspot area and sunspot number obtained by Hathaway [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 Danish
astronomer Christian Horrebow in 1770s on the basis of his sunspot observations
in 1761-1769 [Gleissberg, 1952; Vitinskii, 1965]. Unfortunately, his results were
forgotten and the data lost. Later, in 1843, the amateur astronomer Heinrich
Schwabe found that sunspot activity varies cyclically with the period of about
10 years [Schwabe, 1844]. This was the beginning of the continuing study of
cyclic variations of solar activity [Usoskin and Mursula, 2003]. Figure 2.4 shows
the Schwabe cycle in total monthly sunspot group numbers and in the standard
smoothed sunspot numbers. The 11-year Schwabe cycle is the most prominent
cycle in the sunspot series and in most other solar activity parameters. In addition,
many other parameters including heliospheric, geomagnetic, space weather, and
climate parameters also show the 11-yr periodicity.
The background for the 11-year Schwabe cycle is the 22-year Hale magnetic
polarity cycle, which is also known as Hale’s polarity law. Hale [1908] found that
leading spots and trailing spots in a binary sunspot group have opposite magnetic
polarity. And that leading spots in northern and southern hemispheres have also
opposite magnetic polarity. Furthermore, the polarity of sunspot magnetic field
changes in both hemispheres when a new 11-year cycle starts, see Fig. 2.5. This
relates to the reversal of the global magnetic field of the Sun with the period of 22
years. It is often considered that the 11-year Schwabe cycle is the modulus of the
sign-alternating Hale cycle [Sonett, 1983; Bracewell, 1986; Kurths and Ruzmaikin,
1990; de Meyer, 1998; Mininni et al., 2001]. Although the 22-year magnetic cycle is
the 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 changing
polarity. Gnevyshev and Ohl [1948] studied the intensity (total number of sunspots
over a cycle) of solar cycles and showed that solar cycles are coupled in pairs of a
less intensive even-numbered cycle followed by a more intensive odd cycle. This
is called the Gnevyshev-Ohl (G-O) rule or even-odd rule. Using Rz , the G-O rule
works only since cycle 10 and fails for cycle pairs 4-5, 8-9 and 22-23 [Gnevyshev
and Ohl, 1948; Storini and Sykora, 1997]. When using RG , the G-O rule is valid
since the Dalton minimum and, in the reverse order, even before that [Mursula et
al., 2001].
Silverman [1992] analyzed the power spectrum of monthly SSN for the period
1868 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 since
circa 700 BC in the East (China, Japan, Korea) and in central Europe at latitudes
between 30◦ and 45◦ and found a period of 66 years. The three-cycle quasiperiodicity also exists in a shallow water core taken from the Ionian Sea (Gallipoli
Terrace) [Castagnoli et al., 1997]. Ahluwalia [1998] analyzed the Ap index and
sunspot numbers in the last 6 decades, and pointed out that there is a three-cycle
quasi-periodicity in both data. Ahluwalia [2000] further indicated the existence of
three-cycle quasi-periodicity in the interplanetary magnetic field intensity, and the
solar wind bulk velocity in 1963-1998. Currie [1973] found a period of 66.7-yr in
monthly sunspot numbers in 1749-1957 by using the maximum entropy method.
24
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 the
size of cycle 23 (119). However, all these results are criticized by Wilson and
Hathaway [1999], who claimed that they are mainly due to the short period and
poor quality of data.
The long-term variation of the amplitude of Schwabe cycles is known as the
Gleissberg cycle [Gleissberg, 1944]. However, the Gleissberg cycle is not a cycle
in a strict sense but rather a modulation of the cycle envelope with a varying
time 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 variation
in tree-ring radiocarbon records. Even longer (super-secular) cycles are found in
cosmogenic isotope data. Most prominent cycles are the 600-700-yr cycle and the
2000-2400-yr cycle [Usoskin et al., 2004]. Gleissberg [1952] also suggested a 1000-yr
variation.
2.1.5. Grand minima and maxima
The regular cyclicity of solar activity is sometimes intervened by periods of greatly
depressed activity called grand minima. The last grand minimum and the only
one covered by direct solar observations was the famous Maunder minimum during
1645-1715 [Eddy, 1976; 1983]. There are other grand minima found in the past
from cosmogenic isotope data in terrestrial archives like 14 C and 10 Be, including
Spörer minimum (1420-1530), Wolf minimum (1280-1340), Oört minimum (10101050), Dalton minimum (1790-1830) etc [Eddy, 1983; Kremliovsky, 1994; Komitov
and Kaftan, 2004]. During the Dalton minimum, however, sunspot activity was
not completely suppressed and still showed the Schwabe cyclicity. Schüssler et al.
[1997] suggested that this can be a separate, intermediate state of the dynamo
between the grand minimum and normal activity.
Grand minima and maxima are an enigma for the solar dynamo theory. It is
intensely debated what is the mode of the solar dynamo during such periods and
what causes such minima and maxima [Feynman and Gabriel, 1990; Sokoloff and
Nesme-Ribes, 1994; Schmitt et al., 1996]. A large amount of work suggests that
they 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 improvement
in accelerator mass spectrometry, solar activity can be reconstructed over multiple
millennia from concentrations of cosmogenic isotopes. Accordingly, several grand
minima and maxima in the past were found from there, e.g., Eddy [1977a]; Voss
et al. [1996]; Goslar [2003]; McCracken et al. [2004]; Usoskin et al. [2006]; Brajša
et al. [2009]; Hanslmeier and Brajša [2010]. Stuiver and Braziunas [1989] studied
the 14 C record and suggested two types of grand minima: shorter Maunder-type
and longer Spörer-like minima [Stuiver et al., 1991]. The production of cosmogenic
isotopes 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 of
grand minima in the long sunspot number series [Usoskin et al., 2007a].
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
year
of center
1680
1470
1305
1040
685
-360
-765
-1390
-2860
-3335
-3500
-3625
-3940
-4225
-4325
-5260
-5460
-5620
-5710
-5985
-6215
-6400
-7035
-7305
-7515
-8215
-9165
duration
(years)
80
160
70
60
70
60
90
40
60
70
40
50
60
30
50
140
60
40
20
30
30
80
50
30
150
110
150
comment
Maunder
Spörer
Wolf
a)
b)
a, b, c)
a, b, c)
b, e)
a, c)
a, b, c)
a, b, c)
a, b)
a, c)
c)
a, c)
a, b)
c)
c)
a, c)
c, d)
a, c, d)
a, c)
c)
a, c)
-
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 of
grand maxima in the long sunspot number series [Usoskin et al., 2007a].
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
year
of center
1960
-445
-1790
-2070
-2240
-2520
-3145
-6125
-6530
-6740
-6865
-7215
-7660
-7780
-7850
-8030
-8350
-8915
-9375
duration
(years)
80
40
20
40
20
20
30
20
20
100
50
30
80
20
20
50
70
190
130
comment
modern, b)
a)
b)
a)
a)
-
a) Discussed in Eddy (1977a, b).
b) Center and duration of the modern maximum are preliminary since it is still ongoing.
28
modulated by the geomagnetic field. The earlier reconstruction of solar activity
did not take the magnetic field into account.
Recently, Beer et al [2003] and Solanki et al. [2004] developed a more appropriate reconstruction model by taking both effects into account [see also Usoskin
et al, 2003; Usoskin et al., 2004]. Using this new model, the solar activity can be
quantitively reconstructed. Accordingly, the grand minimum can be quantitively
defined [Usoskin et al., 2006; 2007a] . Usoskin et al. [2007a] defined the grand
minimum as a period when the sunspot number (smoothed with the Gleissberg
filter) is less than 15, while during grand maximum, it exceeds 50 during at least
two consecutive decades. Tables 2.1 and 2.2 show the grand minima and maxima
since 9500 BC found by Usoskin et al. [2007a]. The new definition leads to longer
durations of grand minima. For example, the duration of Spörer minimum in Table
2.1 is 50 years longer than that found by Eddy [1983].
Recent studies suggest that the Sun was more active in 1930s-2000 (cycles 1723) than in other times during last thousand years. Accordingly, this time period
is called Modern Grand Maximum (MGM). However, more recent measurements
have shown that sunspot levels in the declining phase of cycle 23 and in the starting phase of cycle 24 are far below average [Livingston et al., 2012; Penn and
Livingston, 2011]. The last minimum before present sunspot cycle 24 lasted much
longer than earlier minima showing the Sun’s unusual low activity [Li, 2009a]. The
mutual relation between sunspots and solar radiation indices, such as UV/EUV
indices, the 10.7cm flux, has dramatically changed at the end of 2001 after remaining stable for about 25 years until 2000 [Lukianova and Mursula, 2011; Tapping
and Valdés, 2011]. In addition, the average magnetic field strength in sunspot
umbra 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 in
1859 in white (visible) light as a localized brightening of small areas within a
complex sunspot group. It was also the first documented white light flare.
A solar flare is observationally defined as a brightening of any emissions of the
electromagnetic spectrum occurring at a time scale of minutes [Benz, 2008]. Solar
flares are one of the two most important eruptive phenomena in the atmosphere of
the Sun (the other are CMEs). They are the most efficient mechanism to release
magnetic energy associated with active region filaments (prominences), sunspots
or 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 solar
flares can be distributed in many forms: hard electromagnetic radiation (γ- and
X-rays), energetic particles (protons and electrons), and mass flow.
Small flares (microflares, nanoflares and X-ray bright points) can occur all over
the 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 of
small flares can be diagnosed by EUV and X-ray images. Large (or regular) flares
normally occur in active regions with complex magnetic structures observed in
photospheric magnetograms. Flare prolific sites often show signatures of strongly
twisted or sheared magnetic fields. Only regular flares are discussed in the following
sections 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 are
also 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 emission
is denoted by F, N, and B representing faint, normal, and bright. Accordingly, an
SF flare belongs to the smallest class and has a faint brightness, while 4B denotes
the largest class and is bright. Figure 2.6 presents the solar atmosphere at the
flare site in the wavelength band of the Hα line emission observed on 7 August
1972 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 bright
lines 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
A
B
C
M
X
W/m2
I < 10−7
−7
10 ≤ I < 10−6
10−6 ≤ I < 10−5
10−5 ≤ I < 10−4
10−4 ≤ I
Ergs/cm2 /s
I < 10−4
−4
10 ≤ I < 10−3
10−3 ≤ I < 10−2
10−2 ≤ I < 10−1
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 hot
40-million-degree shell intersects the surface at the ribbons and condenses at the
loop tops.
Recently, solar flares are usually characterized and classified by their brightness
in X-ray radiation since the continuous X-ray flux observations by Geostationary
Operational Environmental Satellites (GOES) started in mid-1970s. The X-ray
flux is measured in two bands, 0.5-4.0 Å and 1.0-8.0 Å. X-ray flares are classified
to 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 Å band,
representing typical background levels around minimum and maximum of a solar
cycle, respectively. C, M, and X classes are defined according to the X-ray flux
in 1.0-8.0 Å band, indicating increasing levels of flaring activity. Accordingly, the
biggest flares are called X-class flares. Within these main classes the flares are
further divided into subclasses. For instance, a C5 flare has the measured peak
intensity of 5.0×10−6 W/m2 and an X3 flare 3.0×10−4 W/m2 . Figure 2.7 gives
an example of GOES X-ray flux observation in three days by GOES 8 and GOES
10. GOES 8 (1994-2004) and GOES 10 (1997-2009) both have 0.5-4.0 Å band
and 1.0-8.0 Å band. In Fig. 2.7 the X-ray flux observed by GOES 10 is overlaid
by GOES 8 with perfect matching each other. A large impulsive increase on 14
July 2000 was recorded in the 1.0-8.0 Å 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 and
many C-flares were also observed in the three days. These C-, M- and X-flares
caused also the increases in the 0.5-4.0 Å 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 of
electromagnetic radiation as presented in Fig. 2.8. At first, soft X-rays and EUV
are observed to increase gradually due to the slow heating of coronal plasma in
the flare region. This period is called the pre-flare phase followed by the impulsive
phase, in which most of the energy is released and a large number of energetic
particles (electrons and ions) is accelerated. An evident impulsive increase can be
observed in hard X-rays, γ-rays, and microwaves (millimeter to centimeter radio
waves), and a few impulses can be seen in decimetric and metric radio waves and
EUV. A significant rapid increase is observed in soft X-rays. The fast increase in
Hα intensity and line width terms the flash phase, which coincides partly with the
impulsive phase. After the flash phase, plasma in the low corona (around the flare
region) returns nearly to its original state, while the plasma ejections in the high
corona, the magnetic reconfigurations, and shock waves keep accelerating particles
continuously, causing metric radio bursts and interplanetary particle events. This
period is named the decay phase. Noting that the fluxes of soft X-rays and Hα
have different temporal evolution, flare phases in soft X-rays are characterized as
early 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 flares
in 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 corresponded to GOES soft X-ray classes C9.6 to X17.0 [Ishikawa et al., 2011]. The
space-based observations have provided many striking and excellent images. One
important finding is that the coronal sources of hard X-rays and γ-rays are above
the loop-top, showing systematically different signatures from the double footpoint
sources.
Krucker et al. [2008] present a good example of coronal hard X-ray source
together with two footpoint X-ray sources as shown in the left panel in Fig. 2.9.
This figure shows a TRACE 1600 Å image taken at 06:45:11 UTC, 20 January
2005, overplotted with 12-15 keV red contours (in the corona) and 250-500 keV
blue contours at the two footpoints, implying that the coronal hard X-ray emissions
are fainter than the emissions of the footpoint sources. This made it so hard to
detect the coronal source earlier. The high coronal hard X-ray and γ-ray emissions
show a fast rise and a slower constant exponential decay and the corresponding
Y (arcsecs)
33
X (arcsecs)
0020
0040
Fig. 2.9. Flare observations in hard X-ray and γ-rays. Left: two footpoint sources in γrays 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 the
emissions from the footpoint sources. The right panel in Fig. 2.9 presents the
RHESSI hard X-ray and γ-ray count rates for the 23 July 2002 flare together with
GOES soft X-rays on the top [Lin et al., 2003]. The temporal evolution of hard
X-rays and γ-rays determines three phases- rise phase, impulsive phase, and late
phase. The impulsive phase overlap partly with the impulsive and gradual phases
in soft X-rays. During the rise phase the hard X-ray emission comes from the
coronal source appearing at the top of the flare loop. During the impulsive phase
the emission mainly comes from footpoints, while an additional coronal source
becomes visible in the late phase.
2.2.3. Radio emissions during solar flares
Among prominent characteristics of flares are the large number of line emissions
and the consequences of the rapid energy release to particles. For example, 331
emission lines within the wavelength 3590-3990 Å were observed in the 3B flare of
September 19, 1979 [Fang et al., 1991]. In a flare electrons and nuclei are accelerated nearly to the speed of light. They can produce electromagnetic radiation
throughout the spectrum from radio waves to X- and γ-rays, in which visible lines
of Hα and Ca II H and K are among the most prominent emissions. Usually flares
produce 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 bursts
associated with solar flares are classified in five types according to their dynamical
spectral 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 few
MHz) bursts superposed on an increased continuum. This type of bursts can last
for hours to days caused by supra thermal electrons being accelerated repeatedly
in active regions. Radio noise storms can be observed in association with flares and
CMEs. 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 longer
time. The typical frequency drifts of Type II and Type III bursts are 0.25 and
20 MHz/s, respectively. In normal cases, bursts drift from high to low frequencies
due to the upward motion of electrons along magnetic field lines. The accelerated
electrons 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 of
electrons. Accordingly such bursts are called reverse type III, including ‘U’ shape
and ‘J’ shape [Pick and Vilmer, 2008]. Type II bursts occur a few minutes after
the start of a flare, around the flash phase, and last for more than 10 min. Type
III bursts start to be detectable already a few minutes before the flare impulsive
phase.
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 of
type IV bursts causing a sudden geomagnetic storm is a function of their radio
importance (The radio importance of a flare is defined as the maximum flux density
at 10 cm multiplied by the duration).
Type V radio bursts are a broadband diffuse continuum with the highest frequencies at the start around 150 MHz. They last from tens of seconds to a few
minutes. In general, type V bursts follow type III and were suggested to be caused
by emissions of relatively low energetic electrons at the harmonic of local plasma
in a open magnetic field.
It should be noted that recent radio observations with high resolution show a
few 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 spectrometers of NAOC (National Astronomical Observatories, China). The brightness
temperatures of spikes were suggested to be 1015 K and even higher. Moreover,
submillimeter radio waves at a few hundred GHz caused by ultra relativistic electrons were also detected recently, together with hard X-rays and γ-rays [Raulin et
al., 2004].
2.2.4. Flare models
A classical flare model is the ‘CSHKP’ (Carmichael-Sturrock-Hirayama-KoppPneuman) model. In this model, a current sheet is produced above a closed
magnetic loop before or after magnetic field line reconnection, forming a inverted
Y type cusp-shaped magnetic field structure. Magnetic reconnection is the keypoint in this model. Although this model has been extended by many authors, no
substantial improvement on this model has been achieved so far. The magnetic
reconnection model was even questioned by Akasofu [1995]. However, with the advent of space-borne X-ray image telescopes, more and more evidence for magnetic
reconnection have been obtained since 1990s.
Using X-ray images observed by Yohkoh, Shibata [1999] developed a plasmoidinduced magnetic reconnection model, in which the plasmoid ejection triggers the
impulsive phase. The sketch of this model presented by Shibata [1999] is shown
in Fig. 2.11. The plasmoid (filament/prominence held by a twisted flux rope) is
situated in the current sheet existing in the corona before a flare and prevents the
magnetic field lines from reconnecting before a flare occurs. If the plasmoid starts
to rise, the plasma surrounding the current sheet will fill in the region left behind
the plasmoid, forming an inflow and driving the sheared or anti-parallel magnetic
field lines to reconnect and release energy very fast. Magnetic reconnection further
accelerates the plasmoid until the plasmoid is ejected away from the Sun. What
drives the plasmoid to move originally is not clear. It seems that the structure of
this process is beyond our observing ability at the moment [Hudson, 2011].
A large amount of plasma in the reconnection region is accelerated to energetic (non-thermal) particles, electrons and ions. The downward ejected energetic
37
140
Y (arcsecs)
180
220
Fig. 2.11. A sketch of the plasmoid-induced-reconnection flare model [Shibata, 1999].
900
940
X (arcsecs)
980
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 loop
top can be observed as downward flow in Hα in the transition region and chromosphere. The energetic particles continue propagating downward along the newly
reconnected magnetic field lines, emitting HXR at the footpoints of coronal magnetic loop in the chromosphere where density is high enough for the high-energy
particles to have Coulomb collisions and slow down until reaching thermal speeds.
In some cases, γ-rays and neutrons are observed resulting from the downward
propagation of energetic ions from the accelerated site into the chromosphere.
The heated plasma at the footpoints expands upwards along the magnetic field
loop, filling up the existing coronal loop, observed as “evaporation” of chromospheric material in soft X-ray images. At the footpoints, enhanced Hα and EUV
radiation is also observed. The two footpoints resulting from many reconnected
magnetic lines form two bright ribbons. With the more and more magnetic fields
reconnecting in the outer layers the two flare ribbons appear to move apart gradually.
The energy deposited in the upper chromosphere is transported into the lower
chromosphere and upper photosphere through thermal radiation and conduction.
In some large flare cases, significant increase in white light continuum can be
observed from chromosphere and photosphere during the impulsive phase, and is
highly correlated with hard X-rays [Matthews et al., 2003]. It is pointed out that
white light should be detected from all flares if the sensitivity of telescopes is high
enough [Hudson et al., 2006].
The motion of the plasmoid may cause waves and fast shocks, which can further
accelerate particles and cause solar energetic particle events (SEPs). The acceleration of SEPs is discussed more closely in section 2.4 since most SEP related flares
have accompanying CMEs and large SEPs are considered more likely to be accelerated 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-parallel
magnetic fields and released by magnetic reconnection. The above discussed magnetic reconnection model describes the observed features of flares very well. The
upward streaming of energetic electrons is detected as Type III radio bursts drifting from high to low frequency due to the decreasing density of corona. Reverse
Type III bursts from downward streaming of electrons are occasionally detected
as well [Aschwanden et al., 1995]. The deduced acceleration sites of electrons lie
above the X-ray loops. Hard X-ray sources at the footpoints and the loop top have
been observed by RHESSI in several events (Battaglia and Benz [2006]; see the
left panel of Fig. 2.12). A coronal hard X-ray source above the loop top before the
impulsive 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 observed
with 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 magnetic
reconnection 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 flare
is generally considered as a coronal phenomenon with many chromospheric features observed when the energy is transported into the dense chromosphere. But
where the energy comes from and what triggers the magnetic reconnection is not
very clear yet. Observations of photospheric magnetic fields show that large-scale
sheared magnetic fields are often detected in an active region before a flare, and
an 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 flux
from 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 with
a large amount of plasma and magnetic fields expanding from the low corona
and ejected into the interplanetary space. They are the largest-scale eruptive
phenomenon in the Sun. It is generally believed that the energy released in a
CME originates from the free magnetic energy stored in the corona, i.e., nonpotential magnetic energy in access of potential magnetic energy. This energy is
released during magnetic reconnection into several energy forms, such as plasma
heating, particle acceleration, waves and kinetic energy of the ejecta.
CMEs are morphologically divided into two classes, narrow CMEs and normal
CMEs, due to their angular width less or greater than ∼10◦ . When a CME with
an 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 whole
solar 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 according
to the observed speed of evolution. It was suggested that impulsive CMEs concur
with solar flares, and gradual CMEs are correlated with eruptive filaments. Later
observations show that CMEs, flares and eruptive filaments do occur together in
many cases, but no clear correlations are found. Therefore, CMEs can not be
simply classified as impulsive or gradual and this classification of CMEs is no
longer in use.
With modern soft X-ray and EUV observations, structures of CMEs can be
observed in more detail, although many processes still remain unknown. Accordingly, CMEs are nowadays classified into jet-like and loop-like classes according to
the structure of magnetic fields (open or connected to solar surface as a loop). It
is generally agreed that a jet-like CME (see the left panel of Fig. 2.13) is caused
by the magnetic reconnection within an emerging flux or between a coronal loop
and the open magnetic fields of a coronal hole, resulting in a compact flare (see the
left panel of Fig. 2.14). This reconnection model is similar to the model of X-ray
jets produced during microflares [Shimojo et al., 1996]. Most of narrow flares show
jet-like eruption.
Loop-like CMEs normally consist of three parts, a closed frontal loop followed
by a dark cavity and a bright core (see the right panel of Fig. 2.13). A sketch
of 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 above
discussed model of solar flares, the present CME model has been developed (see
Fig. 2.14), involving flaring X-ray loops, eruptive filament/prominence forming
the bright core in the cavity, and frontal loop.
The global picture of a loop-like CME is as follows. A twisted or sheared coronal
magnetic 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 field
lines under it to reconnect. The fast magnetic reconnection causes a flare in the
low corona close to solar surface. Meanwhile, the field lines tying to solar surface
and keeping the flux rope from flowing away from the Sun are cut off during
the reconnection. When the flux rope moves outwards very fast, it stretches the
magnetic field lines lying above it to form the frontal loop and causes a shock
wave as well, which can further accelerate particles in the interplanetary space.
The stretching or expanding frontal loop leads to low plasma density behind it
which is seen as a dark cavity in white-light.
Unlike flares, the magnetic reconnection is not always necessary in the eruption
of flux rope during a CME. Many other processes can trigger the flux rope to erupt,
such as loss of equilibrium and MHD instability [Vršnak et al., 1991; Vršnak, 2008].
2.4. Solar energetic particles acceleration
Solar energetic particles (SEPs), mainly protons, and a smaller amount of electrons
and heavy ions, were first noted by Forbush as three unusual cosmic ray increases
[Forbush, 1946]. Later they were detected by ground neutron monitors and diagnosed as ground level events (GLEs; also called ground level enhancements). Only
extreme high-energy particles (∼1GeV) from powerful solar eruptions can cause a
GLE. Statistics show that SEPs occur every four days on an average, while GLEs
occur only once a year [Duggal and Pomerantz, 1971]. GLEs are suggested to be
associated with the impulsive phase of flares [Aschwanden, 2012], while large SEPs
are proposed to be mainly accelerated by CME shocks [Cliver and Ling, 2009].
After a few decades of observations and studies, the SEP events were divided
into two types, impulsive events and gradual (long-duration) events according to
the duration of soft X-rays [Cane et al., 1986; Reames, 1999]. The impulsive SEPs
were thought to relate with flare acceleration processes and have accompanying
Type III radio emissions, high ratio of Fe/O, enhanced 3 He and high charge state
of Fe. The impulsive SEPs probably result from wave-particle interaction caused
by magnetic reconnection during a flare, as mentioned in Sect. 2.2.4. The gradual
SEPs were associcated with CMEs and Type II radio burst events, being accelerated by coronal and interplanetary shocks driven by CMEs. Gradual events have a
lower Fe/O ratio and a lower Fe charge state than impulsive events [Cliver, 2009].
However, recent observations show that ‘impulsive’ characteristics are sometimes observed also during gradual events [Cane et al., 2006]. Considering the
newly 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, for
details]. According to this scheme, SEP events are classified into flare and shock
SEP classes instead of impulsive and gradual classes. Shock class has two subclasses, quasi-perpendicular shocks and quasi-parallel shocks (denoted by Q-perp
and Q-par, respectively in Fig. 2.15). Large (gradual) SEP events with impulsive
characteristics are considered to result from quasi-perpendicular shock accelerated
supra-thermal particles originating from flares. While large SEP events without
impulsive characteristics are quasi-parallel shock accelerated supra-thermal particles 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 to
employ a coordinate reference frame. For this purpose we first need to know the
rotation axis of the Sun. The direction of the rotation axis of the Sun is given
by two rotation elements or angles. One is the angle of inclination i between the
ecliptic plane and the solar equatorial plane. The other is the ecliptic longitude of
the ascending node of the Sun’s equator determined by the angle, in the ecliptic
plane, between the vernal equinox direction and the ascending node, where the
Sun moves to celestial north through the ecliptic plane.
Once the axis of solar rotation is known, the solar latitude ψ, also called the
heliographic latitude, is easily defined as the angular distance from the equator.
Frequently used is also the polar angle (angular distance to the North Pole), or
the 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 surface
like the Sun. Carrington introduced in 1850s a rigidly rotating (14.1844 deg/day,
sidereal) longitudinal frame by observing low latitudinal sunspots. This is the
rotating rate of sunspots close to the latitude of 17◦ -20◦ . The sidereal rotating period of this longitudinal coordinate frame is 25.3800 days and its synodic period is
27.2753 days, which is the so-called Carrington rotation (CR) period. CR number
1 was defined to begin at 21:38:44 UT on 9 November 1853 with longitude 0◦ at
the central meridian of solar disc and increasing westwards to 360◦ . CR number
2105 started on the Christmas Eve in 2010. At the start of each new rotation the
longitude 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 Scheiner
in 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 observations
of sunspots although he shared them with a group of correspondents in England
[The Galileo Project: Thomas Harriot 1560-1621]. Johannes Fabricius published
a book in 1611 named ‘Narration on spots observed on the Sun and their apparent rotation with the Sun’. Unfortunately, the existence of the book cannot be
firmly verified up to the time. Scheiner and Galileo independently published their
observations in 1612. Galileo gave a rotation period of 29.5 days. With improvement of the telescope, Scheiner obtained a rotation period of 27 to 28 days from
the observations in 1609-1625 [Scheiner, 1630]. Scheiner also noted the differential
rotation on the solar surface. His book ‘Rosa Ursina’, published in 1630 presented
evidence that the pass of sunspots across the solar disc took a slightly longer time
at high latitudes than near the equator. However, the heliographic latitude and
longitude reference frame was not established until 1863 by Carrington.
It took 250 years until Carington established, by tracking sunspots, that the
surface solar rotation rate decreases with increasing heliospheric latitude, implying
that 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 heliographic
latitude to be as follows
Ω(ψ) = 14.25 − 2.75sin7/4 ψ.
(3.1)
However, for practical purposes the power 7/4 is quite awkward. A more modern
approach is to expand the rotation rate as
Ω(ψ) = A + Bsin2 ψ + Csin4 ψ,
(3.2)
where A is the equatorial rotation rate. A, B, and C need to be determined from
observation.
In addition to sunspots, many other tracers, e.g., photospheric plasma (Dopplergrams), coronal holes (EUV intensity maps) and surface magnetic fields (magnetograms) have been used to determine the rotation rate of solar surface. Some
examples of values for A, B and C determined from different tracers are shown in
table 3.1. From table 3.1 one can see that: (1) the rotation parameters obtained
for different tracers are not necessarily in agreement with each other; (2) the two
sets of parameters determined by all sunspots are close to each other and so are
the groups and recurrent spots; (3) the sunspot groups and recurrent spots have
a lower value of A than sunspots on an average, indicating that these have been
braked by the slower rotation of the solar plasma; (4) the value of B is exceptionally 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
Howard et al. [1984]
Balthasar et al. [1986]
Howard et al. [1984]
Newton and Nunn [1951]
Scherrer et al. [1980]
Snodgrass [1983]
Wagner [1975]
tracers
Mt. Wilson; all spots
1921-1984
Greenwich; all spots
1874-1976
Mt. Wilson; sunspot groups
1921-1984
Greenwich; recurrent spots
1878-1944
Wilcox; photospheric plasma
1976-1979
Mt. Wilson; surface magnetic
fields 1967-1982
OSO-7; coronal holes
May 1972-Oct. 1973
A
14.522
B
-2.840
C
14.551
14.545
14.393
-2.870
-2.720
-2.946
14.368
-2.690
14.440
-1.980
-1.980
14.366
-2.297
-1.624
14.296
-0.510
-0.580
Table 3.1. Coefficients of Eq. 3.2 describing the differential rotation of the Sun’s
surface (in deg/day) measured from different tracers.
3.3. Internal rotation
Helioseismology has revolutionized the study of differential rotation since we can
now empirically determine the rotation even inside the Sun. By using helioseismology inversion of 5-min oscillation data, GONG (The Global Oscillation Network
Group) found out that the rotation velocity at the poles is 3/4 of that at the
equator (see Fig. 3.1). Differential rotation continues throughout the convection
zone. However, near the base of convection zone, rotation changes its character
and becomes solid-body rotation, which continues in the radiative interior. This
region of sudden change in the rotation rate is the so-called “tachocline” [Basu
and Antia, 2003].
Basu and Antia [2003] have drawn a contour diagram of solar rotation as a
function of depth (see Fig. 3.2) by using solar oscillation data of GONG. From
Figs. 3.1 and 3.2 one can see that angular velocity does not change with latitude or
depth in the underlying radiative core, where the radius is less than 0.68R . This
region rotates almost rigidly and the rate is close to the rate of the convection zone
at 30◦ latitude. The rotation rate of convection zone at some latitudes, such as at
15◦ and 60◦ , does not change with depth, i.e., ∂Ω/∂r ≈ 0. There are two shears
in the rotation rate profile (Fig. 3.1), where the rotation rate increases rapidly
with the decreasing radius, i.e., ∂Ω/∂r < 0. One is close to the solar surface, from
the 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 region
is generally believed to be responsible for driving the solar dynamo. The rotation
rate 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 rotation
Ω(r, ψ):
1
r − rc
Ω(r, ψ) = Ωc + [1 + erf(2
)]{Ω(ψ) − Ωc },
2
d
(3.3)
where Ω(ψ) = A + Bsin2 ψ + Csin4 ψ (Eq. 3.2) is the surface latitudinal differential
rotation and erf(x) is the error function. The best fit values of parameters are set
as
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 the
thickness of the tachocline. This definition characterized the differential rotation
profile of solar-like stars by purely latitudinal differential rotation Ω(ψ) with equatorial 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 minimum
towards the equator (poles, respectively). During cycle maximum faster (slower)
bands exist on the equatorial (poleward) side of mean latitude of activity. It is
believed 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 Å
line of Fe I) at the solar surface carried out at Mt. Wilson Observatory in 19681979. They found belts of faster than average rotation that migrate from midlatitudes to the equator. The migration of the zonal flow bands during the solar
cycle is closely connected to the migration of the magnetic activity belt. The
flow pattern was first detected helioseismically by Kosovichev and Schou [1997] in
early 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 converging
to equator with increasing time, there is another system of zonal flow bands at
high 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 representative depths and latitudes obtained using a two-dimensional (2D) regularized
least-squares helioseismic inversion of GONG data in 1995-2005. Moreover, the
amplitudes of the zonal flows at a certain depth change with latitude. Figure
3.4 [Antia and Basu, 2000] presents 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.
Considering the length of a solar cycle as 11 years [Vorontsov et al., 2002], the
time-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 fixed
at 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 from
the surface [Howard and Labonte, 1980; Ulrich et al., 1988; Schou, 1999] to around
0.9R [Howe et al., 2000a; Antia and Basu, 2000], not continuous to the base of
convection zone or tachocline. With accumulation of more seismic data it has now
become clear that the zonal flow pattern penetrates though most of the convection
zone [Vorontsov et al., 2002; Basu and Antia, 2003; Howe et al., 2005; Antia et
al., 2008]. Recently, Howe et al. [2009] made the first study of the differences
in the zonal-flow pattern between the cycle 22/23 minimum and the cycle 23/24
minimum. They suggested that the extended cycle 23/24 minimum during the
last few years is probably due to the fact that the flow bands were moving more
slowly towards the equator. Antia and Basu [2010] confirmed this difference of
zonal-flow pattern between the two minima [Nandy et al., 2011]. They also found
that the low-latitude prograde band that bifurcated in 2005 merged well before
the cycle 23/24 minimum, which is in contrast to this band’s behavior during the
cycle 22/23 minimum where the merging happened during the minimum. The
length of solar cycle 23 in zonal pattern determined to be 11.7 years by correlating
zonal-flow pattern with itself, i.e., shorter than the length derived from sunspot
numbers (about 12.6 years; Hathaway [2010]). Howe et al. [2000b; 2007] reported
a 1.3-yr oscillation in the equatorial region around r = 0.72R during the rising
phase 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 understood, but since the zonal flow banded structure and the magnetic activity pattern
seem 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 interaction
between magnetic fields and differential rotation [Küker et al., 1996; Covas et al.,
2000; 2004].
51
3.5. Meridional circulation
Meridional circulation means a circular flow throughout the meridional plane including longitudinally averaged north-south and radial velocity components. Unlike differential rotation, meridional circulation is not yet fully constrained by
observation, but a poleward surface flow with the velocity of 10-20 m/s has been
detected by various techniques, such as Doppler measurements [Duvall, 1979; Hathaway, 1996], magnetic tracers [Komm et al., 1993; Hathaway and Rightmire, 2010],
and helioseismic analysis [Giles et al., 1997; Braun and Fan, 1998]. The equatorward return flow below the solar surface has not been observed yet. But it must
exist, 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 the
large-scale velocity patterns on the Sun include not only solar rotation but also
another component. This velocity field which they called the “ears” was found to
be symmetric about the central meridian. Beckers [1978] pointed out that the ears
have a similar form to that expected from a meridional flow velocity. Subsequent
work [Labonte and Howard, 1982; Wöhl and Brajša, 2001] confirmed the meridional flows and found radial flows as well. van Ballegooijen and Choudhuri [1988]
studied a model of meridional circulation and suggested a meridional circular flow
velocity form
ur (r, θ) = u0 (
1
c1
c2
R 2
) (−
+
ξn−
ξ n+k )ξsinm θ[(m+2)cos2 θ−sin2 θ],
r
n + 1 2n + 1
2n + k + 1
(3.5)
uθ (r, θ) = u0 (
R 3
) (−1 + c1 ξ n − c2 ξ n+k )sinm+1 θcosθ,
r
(3.6)
where ξ(r) = ( Rr ) − 1, u0 is a measure of the velocity amplitude, and k and m
are two free parameters which describe the radial and latitudinal dependence of
the velocity, respectively (k > 0 and m ≥ 0). The constants c1 and c2 are chosen
so that ur and uθ vanish at the base of the convection zone:
(2n + 1)(n + k) −n
ξ0 ,
(n + 1)k
(3.7)
(2n + k + 1)n −(n+k)
ξ0
,
(n + 1)k
(3.8)
c1 =
c2 =
where ξ0 = ( Rr0 )−1. Applying this mathematical form to a flux transport dynamo
model Dikpati et al. [2004] obtained the typical meridional circulation flow pattern
depicted in Fig. 3.5. The obtained maximum speed of the meridional flow close
to the surface is around 15 m/s.
However, because the meridional-flow speed is more than an order of magnitude
smaller than other flows at the solar surface, it is difficult to be measured accurately. 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 an
analysis. With improved observational techniques, the meridional flow has been
reliably measured at the solar surface using both magnetic elements as tracers
[Komm et al., 1993; Hathaway and Rightmire, 2010] and Doppler velocities [Hathaway, 1996]. These observations show a general flow towards the poles in both
hemispheres that is roughly antisymmetric about the equator. The amplitude of
this flow shows a variation over the solar cycle with fast flows (11.5 m/s) at minimum and slow flows (8.5 m/s) at maximum of cycle 23 [Hathaway and Rightmire,
2010].
Recent results [Beck et al, 2002; González Hernández et al, 2008; 2010; Basu and
Antia, 2010] imply that the pattern of the meridional flow also shows bands of fast
and 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 the
sunspots and the region of low-speed meridional flows and between the migrating
pattern 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 that
the 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 on
the solar surface [Wang et al., 1989] and towards the equator near the base of the
convective zone [Choudhuri et al., 1995; Dikpati and Charbonneau, 1999]. They
also play a role in the generation of differential rotation and its temporal variation
[Glatzmaier and Gilman, 1982; Spruit, 2003] and in the spatiotemporal distribution of sunspots and the solar cycle period [Choudhuri et al., 2007; Nandy and
Choudhuri, 2002; Hathaway et al., 2003]. Variations in the equatorward meridional flow during a solar cycle have been suggested to be able to cause a deep
solar minimum [Nandy et al., 2011]. In the surface flux transport dynamos, the
prolonged cycle 23/24 minimum can be explained by faster meridional flow and
the observed weaker polar fields compared to cycle 22/23 minimum [Schrijver and
Liu, 2008; Wang et al., 2005; 2009], but the prolonged minimum conflicts with
the flux transport dynamos which produce stronger polar fields and a short cycle
with fast meridional flow [Dikpati et al., 2006; Choudhuri et al., 2007]. A fast
meridional flow in the surface flux models can slow down the process of opposite
polarities canceling each other across the equator. Therefore, the flow carries excess of elements of both polarities to the poles with a only slight excess of elements
with the polarity of the polarward side of the sunspot latitudes. This requires a
longer time to reverse the old polar fields and builds up weaker polar fields of the
opposite polarity.
4. Active latitudes
Sunspots are known to appear preferably in a narrow high latitude belt at the
beginning 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 latitude of the emerging sunspots and, more recently, by the magnetic polarity of
sunspots as well. Sunspots of the old cycle and the new cycle coexist during a
few years around solar minima. The latitude migration and the partial overlap
of the successive sunspot cycles leads to the famous pattern of sunspots called
Spörer’s law of zones by Maunder [1903] and was illustrated by his famous butterfly diagram [Maunder, 1904]. NASA’s Marshall Space Flight Center updates
their butterfly diagram together with average daily sunspot area every month at
http://solarscience.msfc.nasa.gov/SunspotCycle.shtml, which shows the latitudinal positions of sunspots for each rotation since 1874. Figure 4.1 shows an example
of their butterfly diagram for the years 1874-2011 and the corresponding sunspot
areas for the same time interval.
Hathaway et al. [2003] investigated the latitude positions of sunspot areas as
function of time relative to the time of minimum.The centroid positions of sunspot
areas in each hemisphere are calculated for each solar rotation. They also studied
the area weighted averages of these positions in 6-month intervals and found that
all cycles start with sunspot zones centered at about 25◦ from the equator. The
equatorward drift is more rapid early in the cycle, slows down later in the cycle
and eventually stops at about 7◦ from the equator [Hathaway, 2010].
Sunspots appear very often in pairs. The trailing sunspot of the pair tends
to appear further out from the equator than the leading spot. Thus the line
connecting 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 the
connecting line to the equator. This pattern of sunspot emergence is called Joy’s
law.
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 towards the poles where they eventually reverse the polar field at the time of sunspot
cycle maximum. The radial magnetic field averaged over longitude for each solar
rotation 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 of
higher 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 uniformly
at 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 Schüssler, 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; Takalo
and Mursula, 2002; Mursula and Hiltula, 2004], total solar irradiance [Mordvinov and Willson, 2003], EUV radiation [Benevolenskaya et al., 2002], and surface helicity [Bao and Zhang, 1998; Pevtsov and Canfield, 1999; Kuzanyan et al.,
2000] are non-axisymmetric and mainly occur in preferred longitude bands. These
preferred active longitudes were given different names when they were found in
different indices, e.g., active sources [Bumba and Howard, 1969], Bartels active
longitudes [Bumba and Obridko, 1969], streets of activity [Stanek, 1972], sunspot
nests [Casenmiller et al., 1986], hot spots [Bai, 1990], helicity nests [Pevtsov and
Canfield, 1999], and complexes of activity [Benevolenskaya et al., 1999].
Even in the very early days of solar physics Carrington [1863] suspected that
sunspots do not form randomly over solar longitude but appear in certain longitudinal zones of enhanced solar activity. However, studies on this topic have been
carried out since 1930s [Chidambara Aiyar, 1932; Losh, 1939; Lopez Arroyo, 1961;
Bumba and Howard, 1965; Warwick, 1954; 1965; Balthasar and Schüssler, 1984;
Wilkinson, 1991]. As a result, the tendency for active regions to cluster in preferred heliographic longitudes on time scales of days, months or even longer were
recognized long.
The longitudinal distribution of sunspots and flares was initially studied in solar 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 [Chidambara 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 appearance 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 reference frames, e.g., in Carrington frame [Warwick, 1965; Vitinskij, 1969; Trotter and
Billings, 1962], in Bartels frame (with synodic rotation period of exactly 27 days)
[Bumba and Howard, 1969; Balthasar and Schüssler, 1983; Bai, 1990], or in other
rotation frame (e.g., 27 13 days in Losh [1939]). Accordingly, active longitudes were
found to appear at different longitudinal positions in different reference frames.
Figure 5.1 shows the longitudinal distribution of first (thick-line) and last (thinline) appearances of two-day spots in solar disc in 1874-1930 [Chidambara Aiyar,
1932]. The frequency of their occurrence suggests a minimum between longitudes
30◦ and 50◦ on either side of the Sun’s central meridian. The maximum frequency
of sunspot occurrence is found to be at different longitudes in reference frames
with different rotation period. Figures 5.2 and 5.3 show the longitudinal distribution of sunspot (Wolf) number in 1903-1937 in 27 13 -rotation-day frame [Losh,
1939] and in 27.2753-day Carrington frame [Warwick, 1965], respectively. The
active longitudes of sunspot activity were found more consistent in 27.2753-day
(Carrington) frame than in 27 13 -rotation-day frame. Warwick [1965] also noticed
that proton flares leading to polar-cap absorption occurred mainly in two active
longitudes separated by around 180◦ .
However, Wilcox and Schatten [1967] pointed out that the Carrington rotation
period is not the rotation period of the Sun, and therefore one should take the
rotation period as a free parameter of the analysis, which needs to be determined by
observations. 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 periodicities
were 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; Özgüç and Atac, 1994].
So-called evolutionary charts of active regions and active-region family trees
are also popular methods to study the longitude of long-lived and complex large
active regions, in which solar flares occur quite often. Dodson and Hedeman
[1968] noticed a band of active longitudes, whose Carrington longitude values
increased in time, suggesting a corresponding rotation period of 27d or slightly
less. 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 and
found two active longitude bands centered around 90◦ and 270◦ in 1970-1971, but
only one active longitude band centered around 180◦ in 1973-1974 (see Fig. 5.4).
Recently, long-lived active longitudes of sunspots with the non-axisymmetry
about 0.1 were suggested by Berdyugina and Usoskin [2003] and the migration
of 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 active
longitude pattern found by Berdyugina and Usoskin [2003] is an artefact of the
used method. In the subsequent work Usoskin et al. [2005] analyzed the effects
of 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 of
randomly generated sunspots is 0.02 and the probability to obtain the 0.1 nonaxisymmetry is less than 10−6 . This implies that the persistent active longitudes
were not an artifact of method. If we correct the observed Carrington longitudes
of sunspots according to the expected migration, the active longitudes should
be consistent in the new frame. Taking this concept into account Usoskin et
al. [2005] found that two persistent active longitudes of sunspots separated by
180◦ have existed during 11 cycles (1878-1996) with the level of non-axisymmetry
about 0.1 in a dynamic reference frame (see Fig. 5.5). If only the position of one
dominant spot for each Carrington rotation is considered the non-axisymmetry
increases to 0.19. This explains why the previously found active longitudes are
not stable in any rigidly rotating reference frame. However, a systematic drift of
active longitudes for sunspot numbers was not found by Balthasar [2007], probably
because the sunspot numbers do not carry good longitudinal information, being
an 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 found
that two active longitudes of X-ray flares have existed during three solar cycles
since GOES started X-ray observations in 1975. They also noted that the level
of non-axisymmetry increases with the X-ray intensity. The non-axisymmetry for
all three classes of X-ray flares was found to be significantly larger than that for
sunspots. 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 strong
X-ray flares occurred in the two preferred active longitudes. Based on the active
longitudes 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 tracers
were found to be quite different; even the parameters obtained for one class of
X-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], the
time resolution for the calculation of the shift of ALs was one Carrington rotation
period. The migration M of ALs with respect to the Carrington reference frame
was given by
Ni
X
M = Tc
(Ωj − Ωc ),
(5.1)
j=N0
where N0 and Ni denote the Carrington rotation numbers of the first and the (i)th
rotation of the data set, Tc = 27.2753 days is the synodic Carrington rotation
period, Ωj is the rotation rate of ALs, and Ωc is the angular velocity of Carrington
frame (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 to
the end of the (i)th Carrington rotation. However, even small difference between
the values of Ωj and Ωc (e.g., 0.2 deg/day) can cause a difference of up to a few
degrees (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 Carrington
rotation 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 rotation
to less than one hour. The time of M is taken the observation time of a flare
or sunspot group transformed to a fractional day k of the Carrington rotation.
Accordingly, the improved version of M has the form
Ni−1
M = Tc
X
(Ωj − Ωc ) + k(ΩNi − Ωc ) ,
(5.2)
j=N0
where Ni−1 denotes the (i-1)th Carrington rotation number. Using this improved
method, we reanalyzed in Paper I the GOES X-ray flare data. The differential
rotation parameters for the three classes of flares were found to agree with each
other and the non-axisymmetry for the longitudinal distribution of flares was systematically improved as well, which gives strong support for the success of the
improved method and the extracted parameter values. The non-axisymmetry of
X-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 the
northern and southern hemispheres rotate at significantly different average angular frequencies during last three solar cycles.
In Paper II, using the improved method we studied the long-term variation of
the differential rotation parameters of active longitudes of sunspots in 1874-2009,
separately in the northern and southern hemispheres. The obtained rotation parameters in the two hemispheres suggest a long-term period of variation of roughly
one century. The difference between the average rotating rates of the northern and
southern hemispheres is in agreement with earlier studies [Bai, 2003a; Hoeksema
and Scherrer, 1987; Brajša et al., 1997; 2000], but does not support the result of
Gigolashvili et al. [2005] who claimed that the northern hemisphere rotates faster
during the even cycles (20 and 22) while the rotation of southern hemisphere dominates in odd ones (cycles 19 and 21). (See Sect. 5.4.2 and Table 5.2 for more
detailed discussion about results on hemispheric difference found in Papers II, III
and 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; Järvinen et al.,
2005a;b; Berdyugina and Järvinen, 2005], and on RS CVn stars [Berdyugina and
Tuominen, 1998; Lanza et al., 1998; Rodonò et al., 2000]. Two permanent active
longitudes 180◦ apart seem to be a conspicuous pattern of stellar activity but can
continuously migrate in the orbital reference frame. The nonlinear migration of
active longitudes is a typical behaviour for single stars, young solar type dwarfs,
and FK Com-type giants, which suggests the presence of differential rotation and
similar 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 other
is the so-called flip-flop phenomenon, which was first observed on the single, rapidly
rotating giant FK Comae [Jetsu et al., 1991]. Doppler images show that the active
regions evolve in size and indicate possible cyclic variations. While one active
longitude reduces its activity level, the other increases. When the active longitudes
have roughly the same activity level, a switch of the dominant activity from one
longitude to the opposite one occurs [Berdyugina et al., 1998; 1999; Korhonen et
al., 2009]. Long time series of photometric data imply that flip-flops are regularly
repeated on FK Comae, RS CVn stars, and young solar analogues, indicating a
flip-flop cycle [Jetsu, 1996; Jetsu et al., 1999; Järvinen et al., 2005b; Berdyugina
and Järvinen, 2005]. A list of stars exhibiting a flip-flop phenomenon was recently
reconstructed [Berdyugina, 2005].
The solar active longitudes also exhibit a flip-flop cycle. An analysis of 120
years of sunspot data [Berdyugina and Usoskin, 2003] suggests that on the Sun
the alternation of the dominant spot activity between the two opposite longitudes
occurs periodically, at about 1.8-1.9-yr period on average, indicating 3.8-yr and
3.65-yr flip-flop cycles in the northern and southern hemispheres, respectively.
Transforming the Carrington longitudes (0◦ -360◦ ) into phases (0-1), they calculated the average phase every half a year, separately for the northern and southern
hemispheres. Accordingly, a phase lag diagram of the two active longitudes with
respect 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 active
longitude delay for about 2.5 rotations per cycle. Filled circles denote the phases
of the active longitude, which is dominant during the corresponding 6 months, and
open circles denote the less active longitude. Curves depict the migration of the
phase lag, resulting from the change of the mean latitude of sunspots and differential rotation. The active longitudes separated by 0.5 (180◦ ) in phase are persistent
in both hemispheres for at least 120 years. The migration continues throughout
the 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 of
dominance to the other active longitude occurs rather rapidly. On an average, six
switches of the activity occur during the 11-yr sunspot cycle and, thus, results in
the 3.7-yr flip-flop cycle, about 1/3 of sunspot cycle.
Similar flip-flop period was also found in heliospheric magnetic field [Takalo and
Mursula, 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 that
there are two persistent active longitudes centered around 90◦ and 270◦ . There are
six groups of flare occurrence in Fig. 5.8 arranged around the two active longitudes.
They demonstrate the periodical alternation of the dominance of active longitudes
in cycle 21. This implies that a flip-flop cycle of flares is 1/3 of sunspot cycle
period as well. A similar 1:3 relation between flip-flop and main activity cycle was
found also in other stars [Berdyugina and Järvinen, 2005].
5.3. Dynamo models
We are interested in the recent studies of the dynamo models for explaining the
active longitudes and flip-flops observed on the Sun and solar-like stars rather
than the conventional solar dynamo theory for explaining the sunspot cycle and
the butterfly diagram.
The typical astrophysical dynamo is modeled by an electrically conducting rotating sphere, whose internal structure and motions are symmetric with respect
both to the rotation axis and to the equatorial plane. A model of this kind excites different dynamo modes that have different symmetries and stabilities in the
kinematic case.
The excited magnetic fields can be decomposed into two parts that are symm
metric (BSm ) and antisymmetric (BA
) about the equatorial plane and have the
following form
m
Bm
A,S = Re[CA,S exp(imφ)],
(5.3)
m
where Cm
A and CS denote the antisymmetric and symmetric complex vector fields
with 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 the
azimuthal coordinate φ (longitude) [Brandenburg et al., 1989; Rädler 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 (picture 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. Assuming 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
1
E=
2µ
Z
B2 × dV,
(5.4)
where the integral is taken over the fluid body and outer space. A measure P of
the degree of symmetry or antisymmetry of a B-field about the equatorial plane
and a measure M for the deviation of a B-field from the symmetry about the axis
of rotation can be defined by
P =
ES − EA
ES + EA
(5.5)
and
M = 1 − E 0 /E,
(5.6)
respectively, where ES and EA are the total symmetric and antisymmetric parts
of E from Eqs. 5.3 and 5.4, E 0 is the energy in the axisymmetric part (M = 0) of
the 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, and
A1 modes, respectively [Rädler et al., 1990; Moss, 2005]. The surface magnetic
fields of the S0, A0, S1, and A1 modes are displayed in Fig. 5.9. In slowly rotating
stars, like the Sun, axisymmetric modes (A0 and S0) are preferably excited. In
more rapidly rotating stars the non-axisymmetric modes (e.g., A1 and S1) get
excited [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 clear
cyclic behaviour and sign changes as in the sunspot cycle, while non-axisymmetric
modes appear to be rather steady. For explaining the flip-flop phenomenon, where
we see both active longitudes and oscillations, an oscillating axisymmetric dynamo
mode 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 a
steady, mixed-polarity non-axisymmetric mode. Moss [2005] studied the nonaxisymmetric magnetic field generation in rapidly rotating late-type stars and
proposed that a combination of S0 and S1 can produce the flip-flop. The superimposed magnetic field structure of S1 and S0 is presented in Fig. 5.10. In this
model if the non-axisymmetric field S1 remains constant and the axisymmetric
component S0 changes its sign and oscillates regularly every half cycle, the two
active longitudes will switch by 180◦ at the same frequency of the S0 field. The
flip-flops occurring at the frequency of the sign changes of the axisymmetric mode
have been observed in some RS CVn stars. More frequent flip-flops, as observed
in single stars and the Sun, suggest a more complex field configuration. Fluri
and Berdyugina [2005] suggested that a combination of one axisymmetric dynamo
mode (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. Fluri
and Berdyugina [2004] showed that flip-flops could also occur due to alternation
of 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, nonaxisymmetric ) [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 the
condition for the excitation of the dominant axisymmetric mode and the condition
that favours the non-axisymmetric mode and developed a coexistence of A0 mode
and S1 mode which results in the flip-flop phenomenon. They pointed out that
the axisymmetric magnetic field is mainly concentrated near the high latitude of
approximately 55◦ at about 0.7R , where the radial shear of differential rotation
is strong, while the non-axisymmetric field occur near the intermediate latitude of
35◦ at the bottom of the tachocline, where the differential rotation is weak and
the diffusivity is low.
The solar dynamo could in principle generate non-axisymmetric magnetic configurations. However, a non-axisymmetric kinematic dynamo mode, even if preferably excited, must rotate rigidly without exhibiting any effect of phase mixing, which results from the effect of the high differential rotation on the nonaxisymmetric field. For a high differential rotation star as the Sun, the relative
rotation between two nearby latitude belts stretches the non-axisymmetric field
and leads to a differential rotation of the non-axisymmetric configuration. This
is known as phase mixing [Bassom et al., 2005; Berdyugina et al., 2006; Usoskin
et al., 2007b]. Phase mixing enhances the losses caused by turbulent diffusion.
Therefore, the non-axisymmetric magnetic structures excited by various dynamos
are expected to rotate rigidly.
A seeming interpretation for the rigidly rotating non-axisymmetric magnetic
field and the differential rotation of active longitudes observed from sunspots was
proposed by Berdyugina et al. [2006] (described in Usoskin et al. [2007b]) in terms
of the so-called the stroboscopic effect. The stroboscopic effect means that the
non-axisymmetric magnetic structure underlying the active longitudes is strongly
affected by differential rotation, while the rotation law for the magnetic structure
is still quasi-solid body. It gives the impression of a differential rotation of active
longitudes.
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 higher
solar 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 to
exist in solar rotation. The variations in solar rotation are suggested to be most
likely related to variations in solar activity [Javaraiah, 2003b; Javaraiah et al.,
2005]. Recently, Javaraiah [2010] proposed that the mean meridional motion of
the sunspot groups exhibits north-south difference as well. We give a brief review
on the N-S asymmetry of solar activity and solar surface differential rotation in
Sect. 5.4.1 and in Sect. 5.4.2, respectively.
75
5.4.1. N-S asymmetry in solar activity
As early as in 1890s Spörer [1889; 1890] and Maunder [1904] noted that there
were often long periods of time when most sunspots were found preferentially in
one hemisphere. Significant N-S asymmetry was found in a large amount of solar
activity indices, such as the sunspot number and areas [Newton and Milsom, 1955;
Waldmeier, 1971; Roy, 1977; Vizoso and Ballester, 1990; Carbonell et al., 1993; Li
et 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 [Ataç and Özgüç, 1996; Joshi and Joshi,
2004], prominences/filaments [Hansen and Hansen, 1975; Vizoso and Ballester,
1987; Joshi, 1995; Verma, 2000; Gigolashvili et al., 2005], photospheric magnetic
fields [Howard, 1974; Rabin et al., 1991; Song et al., 2005], coronal EUV bright
points [Brajša et al., 2005], coronal green line intensity [Özgüç and Ücer, 1987;
Storini and Sykora, 1995; Sykora and Rybák, 2005], and the heliospheric magnetic
fields [Mursula and Hiltula, 2003; 2004; Virtanen and Mursula, 2010]. Thus, the
northern and southern hemispheres should always be considered separately when
studying solar activity. Some results on the preferred hemisphere in cycles 8-23
are displayed in Table 5.1.
However, quantifying the asymmetry itself is problematic. Taking the absolute
asymmetry index produces a strong signal around the solar maximum times, while
taking the relative asymmetry index produces a strong signal around the minimum
times. The absolute asymmetry ∆ and relative (or normalized) asymmetry δ can
be expressed as follows
∆ = An − As ,
(5.7)
An − As
,
An + As
(5.8)
δ=
where An and As denote the activity index in northern and southern hemispheres,
respectively. The high relative asymmetry found during solar minima is due to
only a few sunspot groups appearing on the Sun. If all of them are in the same
hemisphere, the momentary absolute value of the relative asymmetry approaches
to 1 [Swinson et al., 1986; Vizoso and Ballester, 1990].
Table 5.1 shows that during cycle 10, the N-S asymmetry shifts from northern
dominance to the southern dominance and remains there until cycle 13. The N-S
asymmetry shifts back to the northern dominance in cycle 14 and continues to favor
the northern hemisphere until cycle 20. Southern dominance is restored in cycle 21
and persist until recently. This implies that the N-S asymmetry of solar activity
has a long-term characteristic time scale of about 110 yr [Verma, 1993]. Long-term
trend 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, variations
at other periods of about 1 to 4 cycles [Vizoso and Ballester, 1987; 1989; Özgüç and
76
authors
tracer
Bell [1962]
...
Li et al. [2002]
...
...
Bell [1962]
Temmer et al. [2006b]
Li et al. [2002]
...
...
...
...
...
Verma [1987]
...
...
...
...
Temmer et al. [2006b]
Goel and Choudhuri [2009]
sunspot groups
sunspot groups
sunspot groups
sunspot areas
sunspot numbers
sunspot groups
sunspot numbers
sunspot groups
sunspot numbers
sunspot areas
flare index
sudden disappearances
of prominences
major flares
radio bursts
white light flares
radio bursts
HXR bursts, CMEs
sunspot numbers
polar faculae numbers
Table 5.1. Preferred hemispheres in cycles 8-23.
solar
cycle
8-9
10-13
12-13
12-13
12-13
14-18
18-20
19-20
19-20
19-20
19-20
19-20
...
19-20
19-20
21
21
21
21-23
21-23
more active
hemisphere
North
South
South
South
South
North
North
North
North
North
North
North
...
North
North
South
South
South
South
South
77
Ücer, 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 [Özgüç
and Ücer, 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 that
there is a 22-yr period in the N-S asymmetry of solar activity. However, Carbonell
et al. [1993] argued that the 22-yr period in the sunspot N-S asymmetry is not
statistically significant. The N-S asymmetry of solar activity within a solar cycle is
related 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 sunspot
number, X-ray flares, solar flare index, the yearly hemispheric number of active
prominences, the monthly hemispheric sunspot area, the hemispheric averaged
photospheric magnetic flux density and the total hemispheric magnetic flux in
each Carrington rotation. They found that many effects influence the statistical
significance of the N-S asymmetry, e.g., the type of data and data binning. They
proposed different statistical tests for integer, dimensionless data (e.g., the daily
international sunspot numbers, or the daily X-ray flare numbers), non-integer,
dimensionless data (e.g., the hemispheric sunspot numbers), and non-integer and
dimensional data (e.g., flare index, magnetic flux in density). Appropriate data
binning should be considered for different types of activity parameters.
It is believed [Goel and Choudhuri, 2009] that the Babcock-Leighton process
of poloidal field generation is the source of randomness in the solar dynamo and
that the randomness of the Babcock-Leighton process may give rise to a stronger
poloidal field in one hemisphere compared to the other. Feeding the actual value
of the observed polar field at solar minima to a theoretical model based on mean
field equations, Goel and Choudhuri [2009] developed a numerical dynamo model
to explain the N-S asymmetry. They suggested that the asymmetry of the poloidal
field produced at the end of a sunspot cycle is the major factor determining the
asymmetry 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 authors [Gilman and Howard, 1984; Howard et al., 1984; Balthasar et al., 1986;
Hoeksema and Scherrer, 1987; Antonucci et al., 1990; Japaridze and Gigolashvili,
1992; Brajša et al., 1997; 2000; Georgieva and Kirov, 2003; Georgieva et al., 2003;
2005; Mursula and Hiltula, 2004; Javaraiah, 2003a; Nesme-Ribes et al., 1997; Gigolashvili et al., 2003; 2005; 2007; Wöhl et al., 2010].
The measured rotation velocities differ from each other not only for different
types of solar activity formations, but they differ even for the same types as obtained by different authors. The discrepancy of the obtained velocities for different
tracers of solar activity can be explained by their different heights in the solar atmosphere, where the radial differential rotation causes differences. Discrepancies
78
authors
tracer
Javaraiah [2003a]
...
...
Gigolashvili et al. [2003]
...
Scherrer et al. [1987]
Hoeksema and Scherrer [1987]
Antonucci et al. [1990]
Brajša et al. [1997]
...
Brajša et al. [2000]
Knaack et al. [2005]
...
Bai [2003a]
Wöhl et al. [2010]
Paper II
...
...
Paper III
...
sunspot groups
sunspot groups
sunspot groups
Hα filaments
Hα filaments
PMF
coronal magnetic field
PMF
Hα filaments
LBT regions
LBT regions
PMF
PMF
major flares
SBCS
sunspot groups
sunspot groups
sunspot groups
sunspot groups
sunspot groups
solar
cycle
12
13
14-20
20, 22
19, 21
20-21
21
21
21-22
21-22
21-22
21, 23
22
19-23
23
12
13-16
17-23
13-16
17-22
faster
hemisphere
South
North
South
North
South
North
North
North
North
North
North
North
South
North
South
North
South
North
South
North
Table 5.2. Faster rotating hemisphere in cycles 12-23. LBT denotes low-brightnesstemperature, PMF photospheric magnetic fields, and SBCS small bright magnetic
structures.
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 that
the 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 only
for 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, both
the depth of the phenomenon in the solar atmosphere and the time of occurrence
should always be taken into account when studying solar differential rotation and
its variation.
Balthasar et al. [1986] and Balthasar [2007] studied the long-term variation of
solar differential rotation using sunspot groups and sunspot numbers as tracers.
However, they did not separate northern and southern hemispheres. The long-term
variation of differential rotation from cycle to cycle, separated for the northern and
southern hemispheres, has not been widely studied yet. The obtained results so
far are listed in Table 5.2. The results obtained by different authors significantly
contradict each other.
79
In Paper II, we studied the long-term variation of the solar surface rotation
using the improved analysis mentioned in Sect. 5.1 and sunspot observation in
1874-2009. This paper confirms the result [Bai, 2003a; Hoeksema and Scherrer,
1987; Brajša et al., 1997; 2000] that the northern hemisphere rotates faster than
the south in the last five solar cycles and suggests that the long-term period of
N-S asymmetry of solar rotation is around 11 cycles.
Heristchi and Mouradian [2009] and Plyusnina [2010] analyzed the variation of
solar differential rotation in northern and southern hemispheres with time, but
found no N-S asymmetry in cycles 19-23. This is probably due to the tracers they
used. Heristchi and Mouradian [2009] used the daily sunspot numbers, which do
not carry sufficient longitude information since they are smoothed over the whole
solar disc. Similarly, Plyusnina [2010] used the daily values of sunspot group
areas summed over the whole disc and also over each hemisphere. The data was
smoothed in large longitude windows from 30◦ to 90◦ .
Javaraiah [2003a] found that the variation of A (equator rotation angular velocity, see Eq. 3.2) has a 90-yr periodicity and B (latitude gradient) has a 100-yr
periodicity in the northern and southern hemispheres, respectively. Such longterm periodicity of the differential rotation parameters was also found in in the
combination of sunspots in the two hemispheres [Javaraiah et al., 2005]. This suggests a long period of N-S asymmetry of solar rotation in time scale of a hundred
years. The long-term periodicity of the variation of solar surface rotation is generally thought to be related to the long-term trend of solar cycle (i.e., the Gleissberg
cycle), which may result from the coupling of the Sun’s spin and orbital motions
with 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 rotation
have also been proposed [Javaraiah, 2003a;b; Georgieva and Kirov, 2003; Gigolashvili et al., 2005]. Gigolashvili et al. [2005] found that the northern hemisphere
rotates faster during even cycles (20 and 22) while the rotation of southern hemisphere dominates in odd ones (cycles 19 and 21). Changes in the northern and
southern 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 longer
fit intervals, e.g., 3-cycle and 6-cycle intervals, have less uncertainties and more
consistent variation than the parameters obtained for shorter interval such as 1cycle interval (see Table 5 and Fig. 5 in Paper II). The rotation parameters
obtained for sunspots in last three cycles in each hemisphere in Paper II agree
well with those obtained for X-ray flares for the same time period in Paper I. This
gives further support to the improved method.
In Paper III, we reanalyzed the sunspot data in 1874-2009 using 3-cycle running
fit intervals. The results show that the average rotation rates in each hemisphere
vary 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 to
search 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 the
rotation of large sunspots. To study the effect of large spots, the weight in the
merit 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 in
Paper III without weight, and that the long-term variation of rotation in the two
hemispheres agrees well with each other. The faster rotating hemispheres remain
the 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 intervals 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 decreases
with increasing fit interval. This decrease most likely results from the long-term
variation in solar rotation. In Paper IV, we studied the long-term variation in
solar rotation using short fit intervals such as 1-yr, 3-yr and 5-yr. The long-term
variation in rotation parameters obtained for 1-yr, 3-yr and 5-yr intervals also show
the same pattern presented in Fig. 5.11. Figure 5.11 shows the yearly rotation
rate at the average latitude of sunspots of 17◦ in each hemisphere obtained using
3-yr running fit intervals. The rotation parameters obtained for each 3-yr period
are used for the middle year. The yearly parameters are smoothed over a solar
cycle to show the variation pattern more clearly. The long-term evolution of the
rotation in the two hemispheres obtained using such short fit intervals also show an
anti-correlation. Comparing Fig. 5.11 with the right-hand side panels of Figure
5 in Paper II, one can notice that the only disagreement between the rotation
parameters obtained for 3-yr intervals and those obtained for 3-cycle intervals
exists in the first 3-cycle period (1890-1910). This is probably related to the
extremely weak and long solar minima, during which the short interval fits may
easily lose the continuous evolution of phase. During cycles 15-22, the rotation
of the northern hemisphere keeps increasing while the southern hemisphere keeps
decreasing. The results for this period agrees well between short and long fit
intervals.
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 hemispheres
obtained for 3-yr fit intervals is shown in Fig. 5.12. The two peaks of the difference
in the rotation of the two hemispheres at about 1900 and 1990 suggest a long-term
periodicity in N-S asymmetry of about 90 years. This supports the results of
Javaraiah [2003a] who also found a 90-yr periodicity in the differential rotation
parameters.
6. Summary
In this thesis we studied solar active longitudes of X-ray flares and sunspots. We
noted that in earlier work the migration of active longitudes with respect to the
Carrington rotation was based on rather coarse treatment. We improved the accuracy to less than one hour. Accordingly, not only the rotation parameters for
each 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 systematically
enhanced. The non-axisymmetry of X-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. This hightlights the importance of prediction of active longitudes of
solar flares.
Using the improved analysis, the spatial distribution of sunspots in 1876-2008 is
studied. The non-axisymmetry of sunspots is found to be slightly improved. The
statistical evidence for different rotation in the northern and southern hemispheres
is greatly improved by the revised treatment. Moreover, we have given consistent
evidence 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 time
intervals due to the variation of solar rotation. The non-axisymmetry of active
longitudes determined by 3-yr fit intervals is significantly larger than obtained by
11-cycle interval.
In the future work, we will study the correlation between the active longitudes of
sunspots and the active longitudes of flares at short time scales, and we reexamine
the flip-flop phenomenon using the improved analysis.
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