Bachelor Thesis
Simulation of the Solar Cycle based on a
probabilistic Cellular Automaton
Jens Poppenborg
September 2006
Supervised by Prof. Dr. M.-B. Kallenrode
Abstract
English
Today, advanced equipment allows astronomers to see far into space, to observe other
galaxies shortly after their creation. But we don’t have to look that far in order to find
fascinating and unanswered questions. The sun of our own solar system still holds a lot
of mysteries: sunspots, the solar cycle, solar flares and coronal mass ejections - how are
they intertwined, and what has the sun’s magnetic field to do with them?
These phenomena will be described in chapter 2. In the subsequent chapters I will focus
on the topic of this bachelor thesis, the simulation of the solar cycle based on a probabilistic cellular automaton. This simulation, as it is described in chapter 4, eventually
results in a butterfly diagram as well as various other data that will be evaluated in
chapter 5. One of the most important results is the temporal pattern of the release of
magnetic energy during the solar cycle which could be simulated very accurately. Nevertheless, the total energy released in the model exceeds the observed one. Overall, despite
some differences between the observations and the simulation, the results allow a good
representation of the solar cycle, in particular the temporal pattern of energy release.
German
Dank fortschrittlicher Ausrüstung ist es Astronomen heutzutage möglich sehr weit ins
Weltall zu blicken und sogar Galaxien kurz nach ihrer Entstehung zu beobachten. Man
muss jedoch nicht soweit schauen um faszinierende und unbeantwortete Fragen zu finden.
Unsere Sonne zum Beispiel besitzt noch viele dieser Geheimnisse: Sonnenflecken, der Solarzyklus, Sonneneruptionen und Koronale Massenauswürfe - wie hängen sie zusammen,
und was hat das Magnetfeld der Sonne mit ihnen zu tun?
Diese Phänomene werden in Kapitel 2 näher beschrieben. In den folgenden Kapiteln
werde ich dann auf das eigentliche Problem dieser Bachelorarbeit zu sprechen kommen,
die Simulation des Solarzyklus mit Hilfe eines Zellulären Automaten. Diese Simulation,
welche in Kapitel 4 beschrieben wird, liefert neben einem Schmetterlingsdiagramm auch
weitere Ergebnisse welche in Kapitel 5 ausgewertet werden. Eines der wichtigsten Resultate ist die Energieumsetzung während des Solarzyklus welche sehr genau simuliert
werden konnte. Gleichwohl ist die freigegebene Energie in der Simulation größer als die
beobachtete Energieumsetzung. Insgesamt lässt sich jedoch sagen, dass, trotz einiger
Unterschiede zwischen den Beobachtungen und der Simulation, die Resultate eine gute
Repräsentation des Solarzyklus erlauben.
4
Contents
1. Introduction
7
2. Principles
2.1. Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. History of Sunspot Observation . . . . . . . . .
2.1.2. Characteristics of Sunspots . . . . . . . . . . .
2.1.3. Lifetimes and Decay of Sunspots . . . . . . . .
2.1.4. Size Distribution of Sunspots . . . . . . . . . .
2.2. The Sunspot Cycle . . . . . . . . . . . . . . . . . . . .
2.2.1. Variations in the Duration of a Sunspot Cycle .
2.2.2. Spörer’s Law . . . . . . . . . . . . . . . . . . .
2.2.3. Latitude Drift of Sunspots . . . . . . . . . . . .
2.2.4. Relative Sunspot Number . . . . . . . . . . . .
2.3. The Solar Magnetic Cycle . . . . . . . . . . . . . . . .
2.4. The Babcock Model . . . . . . . . . . . . . . . . . . .
2.4.1. Differential Rotation of the Sun . . . . . . . . .
2.4.2. The Convection Zone . . . . . . . . . . . . . .
2.4.3. Active Regions . . . . . . . . . . . . . . . . . .
2.4.4. The Magnetohydrodynamic Dynamo . . . . . .
2.5. Solar Flares . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1. Coronal Mass Ejections . . . . . . . . . . . . .
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3. Methods
3.1. Cellular Automata . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Cellular Automata and Physical Systems . . . . . .
3.1.3. Limitations of Cellular Automata for this Simulation
3.2. The Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Simulation
4.1. Simulation of the Sunspot Cycle . . . . . . . . . . .
4.1.1. Magnetic Flux and the Sun’s Magnetic Field
4.1.2. Generating new Sunspots . . . . . . . . . . .
4.2. The Automaton . . . . . . . . . . . . . . . . . . . . .
4.2.1. Initialisation of a Sunspot Cycle . . . . . . .
4.2.2. Stage 1: Removal of Sunspots . . . . . . . . .
4.2.3. Stage 2: Processing the Remaining Sunspots
4.2.4. Stage 3: Emergence of new Sunspots . . . . .
4.2.5. Stage 4: Computing the Data . . . . . . . . .
4.2.6. Annotations . . . . . . . . . . . . . . . . . . .
4.3. Graphical Representation . . . . . . . . . . . . . . .
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5
Contents
Jens Poppenborg
5. Results
5.1. The Butterfly Diagram . . . . . . . . . . . . .
5.2. Magnetic Field Reversal . . . . . . . . . . . .
5.3. Characteristics of Sunspots in the Simulation
5.3.1. Diameter of Sunspots . . . . . . . . .
5.3.2. Lifetimes of Sunspots . . . . . . . . .
5.3.3. Latitude Drift of Sunspots . . . . . . .
5.3.4. Bipolar and Unipolar Sunspots . . . .
5.3.5. Discussion of Sunspot Characteristics
5.4. Sunspot Cycles in the Simulation . . . . . . .
5.4.1. Number of Sunspots per Cycle . . . .
5.4.2. Sunspot Numbers and Solar Events .
5.4.3. Energy Conversion in Solar Events . .
5.4.4. Total Sunspot Area . . . . . . . . . .
5.4.5. Average Cycle Length . . . . . . . . .
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6. Conclusion
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A. Parameters
49
A.1. The Sun’s Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2. Merging of Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.3. Emergence of new Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . . 51
B. CD-ROM
C. Resources
C.1. Java . . . .
C.2. Matlab . . .
C.3. LATEX . . .
C.4. Vim . . . .
C.5. OpenOffice
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Bibliography
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Acknowledgements
61
6
1. Introduction
Like buried treasures, the outposts of the universe
have beckoned to the adventurous from immemorial times...
George Ellery Hale, 1931
Phillips [2006] reports that, on July 31st 2006, a sunspot pair appeared in the southern hemisphere of the sun and vanished again after a few hours. While this is nothing
unusual, this sunspot was special because it was magnetically backwards. In all other
sunspot pairs in the southern hemisphere at that time the sunspots were ordered north
and south magnetic poles in the direction of the sun’s rotation. For this new sunspot
pair, however, the order of the magnetic poles was reversed. For astronomers and astrophysicists this is a sign that a new sunspot cycle, the 24th since 1765, is approaching.
This special sunspot though was a novelty in many aspects. Most importantly, it appeared at a latitude of 13◦ , which is very unusual for the first sunspots of a new sunspot
cycle. Generally, these can be found at latitudes of about 30◦ north and south of the
equator. Moreover, the sunspot only lasted for a few hours and was not even given a
number by astronomers. In contrast to this, typical sunspots last days or even weeks.
Consequently, scientists felt insecure whether the new sunspot cycle had truly begun.
This doubt has been dispelled on August 25th 2006 when another, larger, backward
sunspot - number 905 - was observed. Unlike the first, this one lasted for several days
before it vanished.
The 24th sunspot cycle itself is expected to be 30% to 50% stronger than the previous
cycle. Moreover, scientists believe that this cycle will be especially stormy, exhibiting a
large number of solar flares and coronal mass ejections (CMEs). In a time where we are
increasingly dependant on electricity, this can have severe influences on our everyday life.
For example, a magnetic storm in the wake of a large CME caused a power failure in the
province of Quebec, Canada, on March 13th 1989 which lasted for nine hours. Also, we
are far more dependant on satellites than we were ever before. The positioning results
of the Global Positioning System (GPS) for instance can be skewed by an increase of
the total electron content (TEC) in the earth’s atmosphere as it is caused by large solar
flares.
These examples not only show why the sun can still fascinate us today, but also why it
is important to understand how the sun works, and more importantly, how such solar
events can be predicted in the future.
This bachelor thesis can - for obvious reasons - not answer such fundamental questions.
Instead, the intention is to test in how far a cellular automaton can be used in order
to simulate the solar cycle and, even more important, the temporal release of magnetic
energy over the course of centuries. The advantage of a cellular automaton over numerical
solutions is that the rules it is based on can be derived from the observations. Numerical
7
1. Introduction
Jens Poppenborg
solutions instead have to be based on mathematical models which are only partially
reliable because the true nature of the sun and more importantly the sun’s magnetic field
is as yet not completely understood. The main problems in such mathematical models
are the actual reversal of the sun’s magnetic field as well as the release of magnetic
energy.
Apart from the general simulation of the solar cycle, one of the most important aspects
is the energy conversion during solar events. These solar events - for example flares
or coronal mass ejections - occur when two sunspots of opposite polarity merge and
neutralise magnetic flux. Unlike what might be expected, the maximum of energy release
must not necessarily occur at the same time as the maximum in sunspot numbers.
Indeed, some of the strongest solar events occur at times when the sun is almost spotless.
This circumstance can be noted in both the simulation as well as in the observations.
The bachelor thesis itself is divided into four chapters. In chapter 2 and chapter 3 I
will introduce the physical principles as well as the computational methods - cellular
automata and their applicability to this problem - respectively. The focus of chapter
2 will be on the observations themselves that have been made about sunspots and the
solar cycle in the past four centuries. Furthermore, the dynamo model by Horace W.
Babcock will give a physical explanation of these observations.
Afterwards, in chapter 4, the simulation itself will be described. Focus here is on the
rules the automaton is based on as well as the relations between these rules and the
observations. Finally, in chapter 5, the results from the simulation will be evaluated.
These results include sunspot characteristics such as the diameter and lifetime distribution but also the number of solar events as well as the energy that was converted during
these events.
8
2. Principles
The sun is new each day.
Heraclitus of Ephesus
The sun, as the Greek philosopher Heraclitus remarked more than two millennia ago, is
new each day. Although his understanding of the sun differs from our own, this sentence
still grasps the essence of what is known today. The hot, largely ionised gas in the
interior of the sun is in constant turmoil, dark spots appear on the sun’s surface and
occasional huge explosions spew out plasma and particles into space. In this chapter I
will describe the phenomenon of sunspots, which are the most obvious sign of the active
sun. While I will focus on the observations themselves in sections 2.1 to 2.3, section 2.4
will deal with a model to explain these discoveries. Last but not least, there will be a
short summary of the phenomenon of solar flares in section 2.5. For a more detailed
introduction of these as well as other phenomenons of the sun I recommend the book by
Lang [1997].
2.1. Sunspots
An easy way to observe the sun is a pinhole camera - two sheets of paper, one of them
with a small hole through which the sun can be projected onto the other paper.1 If
the conditions are good it is possible that small dark spots - sunspots - can be seen in
this projection. Sunspots that can be observed this way are usually very large, having
a diameter of up to 60.000 km [Solanki, 2003]. This is almost five times the diameter of
the Earth, which is 13.000 km. The smallest sunspots are 3.500 km in diameter, which
still is only slightly less than the north to south distance of the European continent at
about 3.800 km.
Figure 1.: Picture of a sunspot on the right side and a pore in the lower left corner. The
small structures are granules, indicating the convective motion of the sun’s
plasma. Image from the Marshall Space Flight Center.
1
Danger: Looking directly at the sun can cause permanent damage to the eye!
9
2. Principles
Jens Poppenborg
Sunspots can be distinguished from the smaller pores by their structure. While sunspots
consist of a dark centre region, the umbra, as well as a brighter outer region, the penumbra, pores consist of the umbra only. This difference can be seen in Figure 1.
The dark colour of sunspots and pores can be explained with their relatively low temperature compared to the surrounding photosphere: while the photosphere has a temperature of about T = 5.800 K, the temperatures of umbra and penumbra are T = 4.800 K
[Pettit and Nicholson, 1930] and T = 5.500 K to T = 5.700 K [Muller, 1973] respectively.
Nevertheless, a sunspot would be brighter than the full moon if it were seen for itself in
the night sky.
2.1.1. History of Sunspot Observation
The earliest confirmed records of sunspot activity by Oriental astronomers date back
to 200 BC [Eddy et al., 1989]. In Christian Europe the existence of sunspots was only
accepted after the invention of the telescope in 1609. Prior to this it was believed
that sunspots were planets or moons orbiting the sun [Casanovas, 1997]. A sunspots
drawing spanning several weeks of observations is displayed in Figure 2. It was drawn
by Christoph Scheiner - a German astronomer and Jesuit - and later published in his
book Rosa Ursina.
Figure 2.: Sunspots drawing from the book Rosa Ursina by Christoph Scheiner. Reproduced from Casanovas [1997].
Today, there exist almost unbroken records for four centuries of sunspot activity. One
of the more prominent features of these records is the Maunder Minimum in the years
10
University of Osnabrück
2. Principles
1645 − 1715 during which the sun has been very quiet and almost no sunspots could be
observed. It was first noticed by Spörer [1887] whose article was later summarised by
Maunder [1890].
2.1.2. Characteristics of Sunspots
A first step towards understanding the nature of sunspots has been made by Hale [1908]
who discovered strong magnetic fields of up to B = 3.000 G in sunspots.2 In comparison
to this, the magnetic field outside sunspots is about B = 1 G while the average magnetic
field on the surface of the earth is only B = 0, 5 G. Hale’s discovery is based upon
the Zeeman-Effect that was first observed by Zeeman [1897], about a decade earlier.
According to this effect, spectral lines split into several components in the presence of
a magnetic field. Shortly after this discovery Hale also observed that sunspots usually
occur in pairs of opposite polarity. These are called bipolar pairs as opposed to the
less common unipolar sunspots. Hale et al. [1919] write that all bipolar pairs consist
of a leading sunspot (in the direction of the sun’s rotation) of the same polarity as the
hemisphere it appeared in as well as a trailing sunspot of the opposite polarity. This is
known as Hale’s polarity law.
Figure 3.: This picture visualises Hale’s Polarity Law as well as Joy’s Law, both of which
describe characteristics of bipolar sunspot pairs. Image from Green and Jones
[2004].
Hale’s polarity law is displayed in Figure 3. Furthermore, it can be seen that the leading
sunspots of bipolar pairs are closer to the equator than the trailing ones. This tilt angle
increases with the latitude: while sunspot pairs within 10◦ of the equator have an average
tilt angle of 2◦ , sunspots pairs at latitudes of 30◦ have a tilt angle of more than 10◦ .
This is called Joy’s law and was first published by Hale et al. [1919].
2
While Gauss is still frequently used, the corresponding SI-Unit is Tesla: 1 G = 10−4 T.
11
2. Principles
Jens Poppenborg
2.1.3. Lifetimes and Decay of Sunspots
The decay rate of sunspots is as of today not known, but either a linear or a quadratic
decay rate are being suggested. Also, depending on different parameters such as their
structure, their position on the sun’s surface and their motion, sunspots can either decay
rapidly or slowly. As for the lifetimes of sunspots, according to Ringnes [1964], more
than 90% of the sunspots decay within less than one month.
2.1.4. Size Distribution of Sunspots
The size distribution of the umbral areas of sunspots is described by a lognormaldistribution of the form
ln
dN
(ln A − ln< A >)2
dN
=−
+ ln
dA
2 · ln σA
dA max
(1)
where < A > is the mean and σA the geometric standard deviation [Bogdan et al., 1988].
Although this relation has only been shown for the umbral areas, it can be assumed that
the same applies for sunspots as a whole. Figure 4 displays the size spectrum as well
as an upper and a lower lognormal fit. The filled circles in the image display very small
sunspots.
Figure 4.: The image displays the size spectrum of umbral areas for 24.615 sunspots
which are marked by crosses. Smaller sunspots are displayed as filled circles.
Furthermore, upper as well as lower lognormal fits have been plotted. Image
from Bogdan et al. [1988].
12
University of Osnabrück
2. Principles
2.2. The Sunspot Cycle
Sunspots, as introduced in section 2.1, do not always appear at the same latitudes on
the sun’s surface. Instead, their latitude of emergence follows a periodicity of averagely
eleven years. This sunspot cycle was first reported by the German astronomer Samuel
Heinrich Schwabe in 1844 [Schwabe, 1844].
Figure 5 displays a butterfly diagram. The first butterfly diagram was drawn by Edward
Walter Maunder in 1904 for the two sunspot cycles from 1874 − 1902 [Maunder, 1904]
and displays the latitudes at which sunspots have been observed during these cycles. In
the top image of a more modern version on a longer timescale the colours indicate the
area of the latitude strips covered by sunspots while the bottom image displays the daily
average area of the sun’s visible hemisphere covered by sunspots.
Figure 5.: The top image shows a butterfly diagram in which the sunspot positions are
plotted for several sunspot cycles. The bottom image displays the daily average
area of the sun’s visible hemisphere that is covered by sunspots for the same
time. Image from the Marshall Space Flight Center.
Some important properties of the sunspot cycle can be derived from the butterfly diagram. At the beginning of a sunspot cycle only very few sunspots can be observed,
mostly at latitudes of about 30◦ north and south of the equator. This time is called
solar minimum. As the cycle advances an increasing number of sunspots emerges until,
5 to 6 years after the cycle started, the solar maximum is reached. New sunspots form
about 15◦ north and south of the equator at this time. In the years following a solar
maximum only few new sunspots emerge ever closer to the equator. Therefore, about
11 years after a sunspot cycle started, the sun is again almost spotless; the next solar
minimum has been reached. It is noteworthy that sunspots of the new sunspot cycle
may already emerge at high latitudes while sunspots of the waning cycle still dissolve at
lower latitudes.
13
2. Principles
Jens Poppenborg
2.2.1. Variations in the Duration of a Sunspot Cycle
Although sunspot cycles usually last about eleven years, it is possible that some cycles
are as short as 7 or as long as 15 years [Kallenrode, 2001]. Furthermore, longer periods of
almost no sunspot activity like the Maunder Minimum (cf. section 2.1.1) or the Spörer
Minimum in the years 1400 − 1510 [Jiang and Xu, 1986] could be inferred from existing
sunspot data or by measuring the amount of carbon-14 in tree rings and beryllium-10 in
ice cores. As a general rule, during periods of high sunspot activity a lower amount of
cosmogenic nuclides such as carbon-14 or beryllium-10 will be created than during times
of low sunspot activity [Solanki et al., 2004].
2.2.2. Spörer’s Law
The previously described migration of the average latitude at which sunspots appear
during a sunspot cycle was first discovered by Richard C. Carrington in 1858. He noticed
that, at the beginning of a sunspot cycle, new spots appear at latitudes between 20◦ to
40◦ while, at the end of a sunspot cycle, they appear about 5◦ north and south of the
equator [Carrington, 1858]. This observation is today known as Spörer’s law.3
2.2.3. Latitude Drift of Sunspots
Tuominen and Kyrolainen [1982] discovered that sunspots at latitudes of more than 20◦
north and south of the equator drift towards the poles while sunspots at lower latitudes
move towards the equator. This is also displayed in Figure 6 which shows the drift rates
of sunspots during the approximate times of solar maximum and solar minimum. The
drift rates at higher latitudes as well as near the equator are - due to limited data sets not reliable.
Figure 6.: The latitude drift of sunspots at different latitudes during solar maximum and
during solar minimum. Image from Tuominen and Kyrolainen [1982].
3
Spörer refined Carrington’s observations in 1861.
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University of Osnabrück
2.2.4. Relative Sunspot Number
In 1849 Rudolf Wolf introduced the relative sunspot number for an easier comparison of
sunspot observations:
R = k · (10 · g + f )
(2)
Here, g is the number of sunspot groups, f is the number of individual sunspots and k is a
normalisation factor based on location and equipment of the observer. The normalisation
factor k was only added by Heinrich Alfred Wolfer, Wolf’s successor at the Swiss Federal
Observatory in Zürich, and chosen to be k = 1 for Wolf’s own equipment as reference.
Using this equation Wolf could reconstruct sunspot data from earlier astronomers as far
back as the cycle from 1755 − 17664 [Wolf, 1859]. From this cycle onward all sunspot
cycles have been numbered consecutively. In Figure 7 the monthly sunspot numbers
starting with this first sunspot cycle have been plotted.
Figure 7.: This image from the Marshall Space Flight Center displays the monthly average sunspot number for all sunspots cycles from 1750 until today.
4
Including the previous solar maximum from 1750.
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Jens Poppenborg
2.3. The Solar Magnetic Cycle
As described in section 2.1.2, the magnetic fields within sunspots were only observed in
1908, more than 60 years after Schwabe discovered the sunspot cycle. Based upon this
discovery, George Ellery Hale not only formulated the polarity law (cf. section 2.1.2) but
also discovered that each sunspot cycle the magnetic field of the sun reverses its polarity
[Hale et al., 1919]. Thus, it takes 22 years for the magnetic field of the sun to restore its
original polarity. This 22-years period is known as the solar magnetic cycle as opposed
to the 11-years sunspot cycle based on sunspot counts only.
Figure 8.: These synoptic charts show the magnetic flux distribution on the surface of
the sun. The top image displays the solar minimum of sunspot cycle 22
which ended in 1996 while the bottom picture shows the solar maximum of
sunspot cycle 23 in the year 2000. Thick black lines mark the neutral line,
dividing the opposing magnetic field polarities. The solid blue lines are isolines of positive, the dashed red ones lines of negative polarity. The isolines are staged B = (0, ±100, ±200, ±500, ±1000, ±2000) µT. Image from
the Wilcox Solar Observatory.
In Figure 8 two synoptic charts from the Wilcox Solar Observatory are displayed. These
maps show the distribution of magnetic flux on the surface of the sun during the time
of solar minimum (top) as well as during solar maximum (bottom). It can be seen that
during solar minimum only a small amount of flux can be found on the surface, usually
spread across large regions, while during solar maximum a large number of small regions
with strong magnetic fields exist, the active regions.
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2. Principles
2.4. The Babcock Model
Until now I have mainly described the various observations related to sunspots and
the solar cycle. In this section I will now introduce the magnetohydrodynamic (MHD)
dynamo model proposed by Horace W. Babcock in 1961. While the true mechanisms of
the sun’s magnetic fields are not yet fully understood, this dynamo model was the first
approach to describe them at least partially and allows a consistent description of the
different aspects of solar activity as they were introduced above. In order to understand
the model several terms have to be explained first.
2.4.1. Differential Rotation of the Sun
Christoph Scheiner remarked in his book Rosa Ursina that the dark spots crossed the
sun with different velocities. He misinterpreted this as proof for his hypothesis, that
sunspots were actually small planets orbiting the sun, which he regarded as a solid star.
Today we know that the sun is not solid but rather consists of plasma which rotates
at different velocities depending on their distance from the equator. The solar rotation
velocity is highest at the equator and decreases towards the poles: one rotation of the sun
lasts about 26 days at the equator and about 31 days at a latitude of 75◦ [Carrington,
1863].
2.4.2. The Convection Zone
The convection zone is the layer situated between the photosphere, which is the visible
surface of the sun, and the radiation zone of the sun (see Figure 9). It consists of hot,
largely ionised gas - plasma, the fourth state of matter - that is in constant motion.
This motion is caused by temperature gradients: hot plasma from the bottom of the
convection zone rises towards the photosphere where it can be seen in the form of granules
(see Figure 1). As the plasma cools down it will sink towards the bottom of the convection
zone again.
Figure 9.: The layers of the sun, labelled from the innermost layer, the core, to the
outermost layer, the corona. The convection zone is situated between the
photosphere and the radiation zone. The distances between the layers are not
drawn to scale.
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2. Principles
Jens Poppenborg
A crucial aspect is the interaction between the plasma and the magnetic field of the sun.
While the sun’s magnetic field is believed to be generated and stored in the tachocline
[Marshall Space Flight Center], a thin layer between the convection zone and the radiation zone, it still influences the motion of the plasma in the convection zone which in turn
twists the magnetic field lines. Nevertheless, the influence of the magnetic field on the
motion of the plasma is very small: the magnetic field can be regarded as frozen-into the
plasma. This condition is caused by the high conductivity of the plasma, which consists
of almost equal amounts of positive and negative particles. A more detailed description
of plasmas and their interaction with magnetic fields is given by Kallenrode [2001].
2.4.3. Active Regions
Regions on the sun’s surface with strong magnetic fields are called active regions. These
active regions always consist of a leading part of the same polarity as the hemisphere they
appeared in as well as a trailing part of the opposite polarity; they obey Hale’s polarity
law as it was described in section 2.1.2. Furthermore, sunspots always spawn within
active regions, but not all active regions spawn sunspots. Whether or not a sunspot will
form depends on the amount of magnetic flux emerging in an active region.
2.4.4. The Magnetohydrodynamic Dynamo
For the following description of the Babcock Model I will rely mainly on the original
article by Babcock [1961]. Additional information acquired from other sources will be
marked accordingly.
At the beginning of a sunspot cycle, during the solar minimum, the sun’s magnetic field
is poloidal. The magnetic field lines run directly from one pole to the other, both inside
as well as outside of the sun (see Figure 10).
Figure 10.: The sun’s magnetic field during solar minimum. Picture (a) shows the magnetic field lines outside the sun, picture (b) those within. In both images it
can be observed that the field lines run directly from one pole to the other,
forming a poloidal field. Image from Green and Jones [2004].
During this time, the only active regions and sunspots that can be observed are residues
of the waning sunspot cycle. They can be found close to the equator, about to dissolve.
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2. Principles
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As the sunspot cycle advances, the magnetic fields stored within the tachocline wind up
due to the sun’s differential rotation. This is known as the Ω − Effect and only possible because the magnetic field lines are frozen-into the plasma. The formerly poloidal
magnetic field of the sun becomes a toroidal field as displayed on the right hand side of
Figure 11.
Figure 11.: The magnetic field lines are frozen-into the plasma of the convection zone and
become wound up as the sunspot cycle advances (picture (c)), thus aligning
themselves almost parallel to the equator (picture (d)). The magnetic field
of the sun is toroidal now. Image from Green and Jones [2004].
While the deformation of the sun’s magnetic field from poloidal to toroidal was already
known to Babcock, a more precise understanding could only be acquired in recent years.
The role of the tachocline layer in this new understanding of the solar dynamo is described
by Gilman [2005].
Apart from the differential rotation, the frozen-in magnetic field lines are twisted by the
convection of the plasma, the twisting being a result of the Coriolis force. This twisting
of the toroidal magnetic field lines - known as the α − Effect - results in separate thick
strands of flux, magnetic flux tubes. The sun’s magnetic field at this time is several times
as strong as the previous poloidal field. For this reason, this stage of the sunspot cycle
was named Amplification by Babcock.
As the flux density within these flux tubes rises, the pressure of the plasma within them
sinks according to the following equation:
pe = pi + pm
(3)
Here, pe is the external gas pressure, pi is the internal gas pressure and
pm =
B2
2 · µ0
(4)
is the magnetic pressure. According to equation 3 the internal gas pressure is always
lower than the external gas pressure of flux tubes. If now the temperature inside and
outside of a flux tubes is the same, the flux tube will start to rise towards the surface.
This effect is named magnetic buoyancy and was first described by Parker [1955].
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2. Principles
Jens Poppenborg
As individual flux tubes reach the photosphere, strands of them will form magnetic flux
loops that break through the surface and form active regions. These active regions are
always bipolar and, due to the alignment of the toroidal field, obey Hale’s polarity law
(cf. section 2.1.2). The emergence of flux loops can be seen in Figure 12. Furthermore,
due to the previously described α − Effect and in particular due to the Coriolis force, the
leading part of an active regions is slightly tilted towards the equator. This is consistent
with Joy’s law (cf. section 2.1.2).
Figure 12.: Picture (e) displays the emergence of flux loops. Active regions form at
the intersection of these flux loops with the photosphere. Image from
Green and Jones [2004].
In Figure 13 the emergence of flux loops from beneath the sun’s surface is displayed.
While the complete region around these flux loops is but one active region, the various
flux loops can each form pores or sunspots, depending on the amount of flux they carry.
Figure 13.: This image displays the emergence of magnetic flux loops from beneath the
sun’s surface. At the intersection of each flux loop with the photosphere
pores or sunspots can form, depending on the magnetic flux the loops carry.
The broad arrows indicate the directions into which the flux loops drift.
Image from Zwaan [1985].
As an increasing amount of flux emerges with these flux loops, first pores and later
sunspots will form. Furthermore, pores and sunspots of the same polarity can merge,5
thus forming larger pores and sunspots.
5
This includes all permutations: pores with pores, pores with sunspots, etc.
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2. Principles
In section 2.1 I have written that sunspots are darker than the surrounding photosphere
because of their lower temperature. Now, this can also be explained. While the granules
around sunspots constantly transport hot plasma towards the surface, sunspots themselves are held on the photosphere for some days up to several months due to the effect of
magnetic buoyancy. This allows for the plasma within magnetic flux loops to be cooler
than the plasma of the surrounding photosphere without sinking towards the bottom of
the convection zone again.
As described in section 2.2.2, sunspots first appear at latitudes of about 30◦ north and
south of the equator and rarely at latitudes higher than 45◦ . Babcock suggests that only
at these latitudes the velocity gradient is large enough to increase the magnetic field
density of flux tubes above the limit required to rise towards the surface. With increasing
time, the magnetic flux density at lower latitudes where the velocity gradients are smaller
increases above the threshold for magnetic buoyancy and new pairs of sunspots form.
At latitudes higher than 30◦ , the magnetic field lines spread upward and merge with the
respective polar cap. As a result of this, sunspots are less likely to appear close to the
polar caps. Furthermore, while new flux tubes rise at lower latitudes, the flux at higher
latitudes eventually dissipates, leaving these regions spotless again.
During the cycle, it is possible that a trailing and a leading part of different active regions
in the same hemisphere merge and neutralise magnetic flux. The released magnetic flux
of such a merging results in a heating of the surrounding photosphere and is often
accompanied by solar flares and coronal mass ejections. These reconnections of sunspots
become more likely as the active regions get closer to the equator. Here, leading parts
of active regions can reconnect across the equator and neutralise each other. The two
remaining parts of these active regions, regardless of whether they have been in the same
hemisphere or not, form a new active region with a flux loop connecting both parts. This
is displayed in Figure 14.
Figure 14.: In picture (f ) two active regions in opposite hemispheres reconnect and neutralise the leading parts of both regions. The trailing parts of both regions
form a new flux loop in picture (g). Image from Green and Jones [2004].
These new active regions now move poleward in both hemispheres. Seeing that the
remaining trailing parts each has the opposite polarity of the initial field of that hemisphere, the magnetic field at the poles will eventually be reversed. Furthermore, Figure
14 already indicates the poloidal field that will be fully formed at the end of a sunspot
cycle. Thus, after averagely eleven years, the sun’s magnetic field is reversed and poloidal
again. A new sunspot cycle - or the second half of a solar magnetic cycle - is about to
begin.
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2. Principles
Jens Poppenborg
2.5. Solar Flares
Carrington [1859] as well as Hodgson [1859] independently observed a very bright appearance on the sun’s surface on September 1st, 1859, which lasted for several minutes
before vanishing again. Due to its strong brightness, this was one of the rare occurrences
where a solar flare could be observed in white light.6 Today, most of these solar flares
are observed using Hα-Filters which isolate the red light emitted from hydrogen atoms
at the sun’s surface. These red spectral lines are intensified by the energy conversion of
a solar flare. For a more detailed introduction of solar flares I suggest the book by Zirin
[1988].
Solar flares themselves are sudden releases of huge amounts of energy of up to 1025 J
in the form of electromagnetic radiation, accompanied by a heating of the surrounding
photosphere. They occur when sunspots of different bipolar pairs and of opposite polarity
merge, forming a δ-spot 7 [Zirin and Liggett, 1987]. Along the neutral line of the sunspots
magnetic flux is cancelled, resulting in a heating of the surrounding photosphere as well
as solar flares.
2.5.1. Coronal Mass Ejections
While solar flares have been known for more than a century, it was only in the 1970s that
another phenomenon - the coronal mass ejection or CME - was observed thanks to the
development of the coronograph. These CMEs consist of energetic particles and plasma
- and thus magnetic fields - that are spewed out into space. Just like solar flares, their
origin is at the neutral line between sunspots of opposite polarity. Their relation to solar
flares, however, is not yet clear. Most flares (about 90%) occur without an accompanying
CME while about 60% of the CMEs occur without a solar flare. However, in the largest
events both aspects of magnetic energy release are present.
6
White light images of the sun can either be produced by using a special filter that eliminates 99.9%
of the light emitted from the sun, or, as in Carrington’s case, by projecting the image of the sun.
7
δ-spots can also form from a complex single active region or by new emerging magnetic flux around
an already existing older sunspot.
22
3. Methods
It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.
Sir Arthur Stanley Eddington
While in the previous chapter I have described the physical principles upon which the
simulation is based, here I will introduce the computational methods. The initial idea
was to use a cellular automaton as described in section 3.1. Due to its limitations though
(cf. section 3.1.3), the actual simulation only partially resembles a cellular automaton
now. Finally, in section 3.2, I will explain why I am using a one-dimensional lattice
instead of a two-dimensional grid, which at first glance appears to be more appropriate
for this problem.
3.1. Cellular Automata
The idea of a cellular automaton was first formulated by John von Neumann in the late
1940s. Von Neumann proposed the model of a machine that was able to reproduce itself
with an identical complexity. While his model was still very complex, having about
200.000 cells and 29 states for each of these cells, later cellular automata were more
simple in their design but still showed a complex behaviour. A good example is the
game of life devised by John Conway. The algorithm is described by Gardner [1970].
3.1.1. Definition
This definition for cellular automata has been adapted from Chopard and Droz [1998].
A cellular automaton consists of an infinite lattice of cells of the dimension d. Each of the
cells ~r in this lattice is in the local state φi (~r, t) at the time t = 0, 1, 2, .... Furthermore,
there are rules R = R1 , R2 , ..., Rm that define the evolution of the state of each cell as
the time t advances. These rules are usually based on the state of all cells in the direct
neighbourhood of a cell. For instance, in Conway’s game of life, the number of the eight
cells8 in the neighbourhood of a cell that are in the state φj (~r, t) = 1 was counted to
calculate the new state of that cell.
In real simulations, instead of an infinite lattice, the system consists of a finite grid for
which boundary conditions are defined. For example, the two-dimensional game of life
uses a periodic boundary condition. This means that the upper and lower sides as well
as the right and left sides of the grid are connected with each other. The resulting
8
As Conway’s game of life uses a 2D world, each cell has 8 neighbours. Generally, the neighbourhood
depends on the specific problem as well as the dimension of the world it is set in.
23
3. Methods
Jens Poppenborg
globe of this boundary condition is also suited for a two-dimensional representation of
the sun’s surface. Furthermore, while the original definition describes a deterministic
cellular automaton, several physical problems entail a certain amount of randomness
(e.g. radioactive decay). Thus, a number of probabilistic cellular automata exist in order
to describe complex phenomenons. As described in chapter 4, the simulation of the solar
cycle also makes use of probabilistic elements.
3.1.2. Cellular Automata and Physical Systems
Many aspects of nature can be described by differential equations. As these equations
get ever more complex their computation and visualisation can either be done by approximate numerical means or, in some cases, by using cellular automata. Several examples
for the use of cellular automata in physics are described by Chopard and Droz [1998]. In
the case of this simulation, a description of the phenomena using differential equations is
not even possible as our understanding of the underlying mechanisms is still incomplete.
Therefore, the simulation is based on observations as they were described in the previous
chapter but not on a mathematical model.
3.1.3. Limitations of Cellular Automata for this Simulation
While cellular automata can be applied easily to the description of the motion of particles in a gas or a fluid, I have encountered several problems whilst programming this
simulation that eventually forced me to choose a different approach. Here, I will only
describe the problems. The implementation of the simulation can be found in chapter 4.
For the simulation I use a two-dimensional grid in which the y-axis represents the latitude
at which sunspots appear on the sun and the x-axis represents the time. Thus, it strictly
is a one-dimensional finite lattice as the x-axis only displays the time evolution. Now,
while each cell in this lattice represents a certain latitude strip, an individual sunspot
must not necessarily be confined to just one cell. Moreover, several sunspots can be
distributed on the surface of the sun in such a way that they can be found at the same
latitude. Establishing rules which not only could keep apart the individual sunspots
as well as the latitudes they appear in but also still obey the definition of a cellular
automaton was not possible. The problem itself is displayed schematically in Figure 15.
Figure 15.: This image displays the problem of individual sunspots occupying several
cells, as well as the problem of several sunspots occupying the same cell.
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3. Methods
Another problem posed the use of the neighbourhood at the time t in order to calculate
the new state of a cell at the time t + 1. For example, the emergence of new sunspots
is only on a larger scale dependant on sunspots in their direct neighbourhood. Also,
sunspots of opposite polarity can reconnect across the equator even if they are not next
to each other (e.g. one cell).
Overall, while the general idea of a cellular automaton to use a certain set of rules,
applied to the state of a system at the time t to calculate the new state at the time t + 1
still applies, the final implementation of the simulation still differs from the definition.
3.2. The Lattice
For the simulation the sun is divided into latitude strips of 0, 5◦ that cover 60◦ north
and south of the equator. The longitude at which sunspots appear is neglected. Overall,
the surface of the sun can be regarded as a one-dimensional array with 241 cells, 120
cells for each hemisphere and one cell for the equator (see Figure 16).
Figure 16.: The surface of the sun as it is represented in this simulation. Only latitudes
of up to 60◦ north and south of the equator are regarded. The longitudes at
which sunspots appear are neglected altogether.
Alternatively, the complete surface of the sun - including both latitude as well as longitude - could have been modeled each time step. This would have enabled a more precise
placement of sunspots on the sun’s surface. Also, several of the rules described in section
4 would not have to rely as much on randomness as they do now. Nevertheless, the following examples are only some of the problems one has to deal with in a two-dimensional
representation of the sun:
• How do sunspots look like (circle, rectangle) and how to represent them in a twodimensional lattice. This is important insofar as each cell that is occupied by a
sunspot in a two-dimensional grid cannot hold a second sunspot. For this reason,
sunspots have to be well-defined. On the other hand, in the one-dimensional grid I
use, each latitude strip can hold several sunspots without additional requirements.
This greatly simplifies the rules required for sunspot emergence as well as sunspot
migration.
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3. Methods
Jens Poppenborg
• Where exactly do new sunspots emerge relative to each other. For instance, a new
sunspot pair is not allowed to emerge between the leading and the trailing sunspot
of another bipolar pair.
• Where and why does energy conversion occur. A more accurate model of the solar
cycle using a two-dimensional grid would also have to deal with these question. For
example, one solar flare can trigger another flare which is known as a sympathetic
flare. Also, emerging flux on one side of the sun can cause a CME on the other side
of the sun (disconnection event). Seeing how little is known about the reasons for
the occurrence of these solar events, a two-dimensional model trying to simulate
them would have to rely on assumptions and probabilities as well.
• If the cells in the simulation were using both latitude as well as longitude, these
cells would be larger near the equator than near the poles. This would cause
problems when sunspots migrate either towards the equator or towards the poles
as their area would either grow or shrink respectively. In order to conserve the
magnetic flux within these sunspots, additional rules would have to be formulated.
Overall, while a two-dimensional representation of the sun would certainly have been
more accurate in some aspects, the accompanying problems would just as likely have not
been possible to solve within the time I had to finish this bachelor’s thesis. Furthermore,
it is likely that, while some of the abstract rules used in the one-dimensional model
might be replaced by more accurate ones, a two-dimensional model would be based on
assumptions and probabilities elsewhere.
26
4. Simulation
Those who can, do; those who can’t, simulate.
Anonymous
After the principles upon which this simulation is based have been introduced in the
previous chapter, I will describe the implementation of the simulation here. In section
4.1, I will explain some basic assumptions and rules of the simulation while the actual
rules of the automaton will be introduced in section 4.2.
4.1. Simulation of the Sunspot Cycle
The simulation creates instances of the class Sunspot and stores them in a LinkedList.
While in a typical cellular automaton the lattice is usually represented directly by an
array, I used the above solution due to the problems listed in section 3.1.3.9 Furthermore,
unlike the real world in which time is continuous, time in this simulation is discrete as it
is in all simulations. One time step in the simulation equals one month in reality. The
same can be found in the butterfly diagram in Figure 5, where Maunder averaged the
number of sunspots at each latitudes strip for one solar rotation.10
4.1.1. Magnetic Flux and the Sun’s Magnetic Field
In order to simulate the reversal of the sun’s magnetic field, the northern and southern
polar caps are initialised with a fixed value for the magnetic flux at the start of the
simulation - one with a positive, the other with a negative polarity. During each sunspot
cycle, this field is first reduced to zero,11 the solar maximum. Afterwards, the initial
value is restored with an inverse polarity until the end of the sunspot cycle. The value
chosen here is important as it determines the length of a sunspot cycle - the sooner the
magnetic field can be reversed, the sooner the new cycle will start.
The magnetic field at the polar caps is reduced by the magnetic flux of dissipating trailing
sunspots in the respective hemispheres. In order to calculate the magnetic flux I made
the assumptions that sunspots are circles and not cambered across the sun’s surface.
With these assumptions the area of a sunspot is
A = π · r2
(5)
9
For similar reasons, this would have to be done in a two-dimensional lattice as well.
A solar rotation seen from the earth lasts about 27 days.
11
The field does not have to reach exactly zero though.
10
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4. Simulation
Jens Poppenborg
Now, the magnetic flux within a sunspot can be calculated using the following equation:
Φ=B·A
(6)
The radius of sunspots is discrete and can only be a multiple of 3.000 km. This is a
result of the lattice, in which each cell represents a 0, 5◦ latitude strip. Given that the
circumference of the sun is about 4.320.000 km, each of these latitude strips has a length
of approximately 6.000 km, which is thus the smallest diameter a sunspot can have in
this simulation. In order to avoid an arithmetic overflow during the simulation I will use
the number of cells divided by two as the radius of sunspots instead of this factor. This
decision also has a positive influence on the simulation, as the reversal of the magnetic
field will be more balanced at both poles.
4.1.2. Generating new Sunspots
During the simulation, sunspots are always created as bipolar pairs consisting of a leading
sunspot of the same polarity as the hemisphere it appeared in as well as a trailing sunspot
of the opposite polarity. Thus, Hale’s polarity law is accounted for. The latitude at
which a sunspot has the largest distance to the equator is called offset and depends on
the current stage of the sunspot cycle during the simulation. Furthermore, the offset of
the leading sunspot is - depending on the latitude of emergence - closer to the equator
than the offset of the trailing sunspot. For example, bipolar pairs which emerge within
10◦ of the equator can have an axis tilt between 0◦ and 4◦ while sunspot pairs which
appear at latitudes of more than 20◦ can have an axis tilt between 8◦ and 12◦ . This
represents Joy’s law.
All sunspots can span from 1 to 20 cells, equalling a diameter between 6.000 km to
120.000 km. In section 2.1.4 I have written that the sunspot size, and thus the radius,
is described by a lognormal-distribution. Seeing that the smallest diameter of sunspots
in this simulation still is as large as 6.000 km, only the rightmost downward slope of the
resulting graph (see Figure 4) can be modeled. As an approximation for this slope I use
an exponential function in order to choose the diameter of both sunspots. Nevertheless,
using an exponential function is only a rough estimate and should only be regarded as
such. After both diameters have been calculated, the larger diameter is assigned to the
leading sunspot.
Finally, the magnetic field is chosen randomly from an uniform distribution between
1500 G to 3000 G (in steps of 100 G) for the trailing sunspot corresponding to the observed typical magnetic field strengths of sunspot umbras. The magnetic field of the
leading sunspot is then calculated according to the following equation:
B1 = −
Φ2
A1
(7)
Here, Φ2 is the magnetic flux of the trailing sunspot, A1 is the area of the leading sunspot
and B1 is the magnetic field of the leading sunspot. This ensures that both sunspots
have the same amount of magnetic flux which is required because both are intersections
of the same magnetic flux tube with the photosphere.
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4. Simulation
According to Brants and Zwaan [1982] larger sunspots have a stronger magnetic field.
In the simulation, this would cause problems as the following scenario will display. Assuming that the trailing sunspot is very small, according to Brants and Zwaan [1982] the
magnetic flux density within this sunspot would also have to be very small. For a very
large leading sunspot, using equation 7, the magnetic flux density within this sunspot
would then have to be even smaller compared to that of the trailing sunspot. As the
observations show, this problem does not occur in reality because sunspot groups instead
of individual sunspots emerge. In these groups, of which the trailing one often consists
of a lot of small sunspots, the magnetic flux can be distributed among several sunspots.
Also, the magnetic flux of sunspots in the simulation is generally larger than it would
be in reality. This is caused by the magnetic field B which is chosen for the complete
sunspot. In reality, the magnetic field is strongest in the centre of a sunspot, the umbra,
and weaker in the penumbra.
4.2. The Automaton
During the simulation, several methods are called sequentially during each time step (t)
in order to calculate the state of the simulation in the next round (t + 1). In this section
I will describe these methods, which are the rules the automaton is based upon.
4.2.1. Initialisation of a Sunspot Cycle
The first sunspot cycle is initialised at the start of the simulation while subsequent cycles
are initialised as soon as one of the following two conditions has been met:
• Either the magnetic field at both polar caps has been reversed and less than 20
sunspots remain on the sun’s surface, or
• the regions where new sunspots emerge is within 2, 5◦ north and south of the
equator and less than 20 sunspots remain on the sun’s surface.
This second condition helps to avoid prolonged low sunspot activity at the end of a
sunspot cycle, but it also prevents the magnetic field from being completely reversed in
some sunspot cycles. During the initialisation itself, a maximum of four bipolar sunspot
pairs can emerge in both hemispheres. While one bipolar pair is always created per
hemisphere, the remaining three pairs emerge with a probability of 2/5 each. Overall,
the change from one sunspot cycle to the next is more abrupt in the simulation than in
reality. This is caused by the very simplified rules for sunspot emergence in the course
of a sunspot cycle (cf. section 4.2.4). Usually, according to the monthly mean American
sunspot numbers12 from the National Geophysical Data Center, between two sunspot
cycles there are several months or even a year during which only a limited number of
sunspots can be observed. Nevertheless, the sunspot count is seldom lower than four
sunspot per month.
For each sunspot pair the offset is calculated by adding up 120 randomly created numbers
(0; 1). As can be seen in Figure 17 this results in a Gaussian distribution at latitudes
12
These include only high quality observations which are more reliable than the relative sunspot number
as it was introduced in section 2.2.4.
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4. Simulation
Jens Poppenborg
of 30◦ north and south of the equator. Alternatively, Java offers a function to create
normally distributed numbers. As the parameters for this function can’t be set manually
though, the numbers are spread too wide for this purpose.
Figure 17.: The image displays the latitude at which the first sunspots of a sunspot
cycle would appear. An overall of 100.000 values has been calculated for
both hemispheres.
Finally, the offset with most distance to the equator for both hemispheres is stored.
These regions, which will gradually migrate towards the equator as the sunspot cycle
advances, are where new sunspots will emerge. Thus, they represent Spörer’s law in the
simulation.
4.2.2. Stage 1: Removal of Sunspots
While sunspot cycles are only initialised if certain conditions are met, the following three
methods are called during each time step. This first method removes sunspots based on
the following conditions.
First of all, sunspots which have a diameter of less than 6.000 km or a weak magnetic
field of less than B = 1000 G dissipate. If the removed sunspot was part of a bipolar pair,
its partner remains as a unipolar sunspot. Furthermore, if the removed sunspot was a
trailing sunspot and thus had the opposite polarity of the hemisphere it appeared in, its
magnetic flux will migrate poleward and work towards the reversal of the sun’s magnetic
field. The magnetic flux of leading sunspots continues to move equatorward where it
will eventually cancel itself with flux of the opposite polarity. This latter process is not
further observed in the simulation.
Secondly, two sunspots of opposite polarity and of different bipolar pairs in the same
hemisphere can interact with each other if they are close enough together (latitude).
In this case, which occurs with a probability of 2%, magnetic flux is cancelled out in a
solar event such as a flare, a CME, or both. Both sunspots will lose the same amount of
magnetic flux. This usually results in one sunspot vanishing altogether, while the other
30
University of Osnabrück
4. Simulation
sunspot loses in size and magnetic field density. The remaining sunspot can, however,
merge with the other sunspot from the bipolar pair the deleted sunspot was a part of.
Last but not least, the above can also happen with a probability of 90% if two sunspots
of opposite polarity are within 5◦ of the equator in different hemispheres. This is the
scenario described by the magnetohydrodynamic dynamo model in section 2.4.4.
The probabilities of 2% and 90% respectively have been chosen because interaction
between sunspots is more likely at the equator than at higher latitudes.
4.2.3. Stage 2: Processing the Remaining Sunspots
In this stage, sunspots drift - depending on their latitude - either towards the equator or
towards the poles. Sunspots at latitudes higher than 25◦ north and south of the equator
drift one cell towards the poles with a probability of 50% and towards the equator
with a probability of 15%. At lower latitudes the chance that the sunspot will move
towards the equator is 70% and 20% that it will move towards the respective pole. This
approximately reproduces the drift rates of sunspots displayed in Figure 6.
Furthermore, all remaining sunspots decay by one cell with a probability of 50% in this
method. This represents a linear decay rate of sunspots. The realeased magnetic flux of
trailing sunspots is used to reverse the magnetic field in the corresponding hemisphere.
4.2.4. Stage 3: Emergence of new Sunspots
Finally, at the end of each time step, new sunspots are created. In section 4.2.1 I have
described that the offset with most distance to the equator has been stored for both
hemispheres during the initialisation of a sunspot cycle. Now, new sunspots appear up
to 15◦ above this latitude. The same can be observed in the butterfly diagram (see
Figure 5) where the first sunspots of a new sunspot cycle emerge at lower latitudes than
the sunspots that appear in the year following the solar minimum. Nevertheless, apart
from this, the latitude of sunspot emergence still gradually migrates towards the equator
as the cycle advances.
The number of new sunspots per round is determined by the already existing number of
sunspots in each hemisphere. Until the solar maximum is reached, new sunspots appear
with a probability of 25% for each existing sunspots. Between solar maximum and solar
minimum, new sunspots only appear with a probability of 23%. Alternatively, a fixed
amount of new sunspots could be created each round as it is done during the initialisation
of new sunspot cycles. This, however, would have further limited the variance in sunspot
counts during different sunspot cycles. The values I have chosen here limit the overall
number of sunspots per cycle to about 2.000 − 5.000.13 Overall, it can be said that
the dissipating magnetic flux, which is responsible for the reversal of the sun’s magnetic
field, indirectly limits the sunspot numbers. The magnetic flux itself cannot be limited,
as a large part of it is released in solar events. This is consistent with Babcock’s dynamo
model: only a fraction of the magnetic flux on the sun’s surface is required for the
reversal of the sun’s magnetic field.
13
An evaluation of these total sunspot numbers per cycle will be given in section 5.4.1.
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Jens Poppenborg
In case of low sunspot activity with less than 20 sunspots in a hemisphere, or if the
magnetic field is reversed faster in one hemisphere than in the other, one additional
bipolar sunspot pair can emerge per round and hemisphere. This not only prevents
that no sunspots emerge in a hemisphere after the initialisation,14 but also balances the
reduction of the magnetic field in both hemispheres. Sunspots created here can emerge
at latitudes of up to 10◦ above the current offset.
Lastly, the offset where new sunspots appear migrates towards the equator. Depending
on the number of sunspots in the corresponding hemisphere, it can migrate up to 1, 5◦
towards the equator. Thus, during cycles of high sunspot activity the equator is usually
reached sooner than during cycles of low sunspot activity, effectively shortening the
sunspot cycle. This, too, is in agreement with the observations.
4.2.5. Stage 4: Computing the Data
After the new state of the simulation has been calculated, the centre regions of all
sunspots are calculated and their diameter is added up for each latitude strip. Depending
on the area, the cell will be coloured in yellow if more than 1% of the latitude strip’s
surface is covered by sunspots, in red if more than 0, 1% is covered and in black if less
than 0, 1% but more than 0% is covered by sunspots.
Overall, an array with 1920 columns is filled during the simulation.15 As each round
equals one month, the simulation covers 160 years. At the end of the simulation, this
array is used to create the butterfly diagram.
4.2.6. Annotations
The order in which these rules are called is only partially exchangeable. For example, it
is not important whether the migration of sunspots or the emergence of new sunspots
is computed first. The removal of sunspots, however, should always be done first as it
influences the functioning of the remaining methods. Finally, it is obvious that the data
can only be processed after all methods have been called. In Appendix A I will exemplarily show the influence different parameters have on the butterfly diagram resulting
from the simulation.
4.3. Graphical Representation
The graphical representation of the butterfly diagram is realised in the class GUI. This
class is an extension of the JPanel and can as such be embedded in either an application
or an applet. After the simulation has finished, first the coordinate system and then the
butterfly diagram is drawn. Due to the amount of data the butterfly diagram is plotted
in two separate coordinate systems, each covering 80 years. Furthermore, an instance of
the class Evaluation is created that stores various results of the simulation in text files.
These results are put into graphs using Matlab and will be discussed in section 5. For
this bachelor thesis I have only embedded the simulation in an application.
14
Otherwise, if all sunspots had been removed very soon after their emergence, it could happen that no
further sunspots emerge in that hemisphere.
15
The rows represent the latitude, the columns represent the time evolution, cf. section 3.2.
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5. Results
It is much easier to make measurements than to
know exactly what you are measuring.
John William Navin Sullivan
The simulation described in the previous chapter does not only yield a butterfly diagram
but also several other results such as the released magnetic flux, various characteristics
of sunspots and, even more important, the release of energy in flares and other solar
events. These results will be visualised and compared to the observations described in
section 2. All results are from the same run of the simulation. This is helpful in pointing
out the relations of the graphs and the butterfly diagram.
5.1. The Butterfly Diagram
The butterfly diagram displayed in Figure 18 is exemplary for the simulation and will be
used to describe similarities as well as differences with the butterfly diagram displayed
in Figure 5.
Figure 18.: This is an exemplary butterfly diagram from the simulation. A comparison
with the butterfly diagram gained from sunspot data is given in the text.
Overall, the butterfly diagram in Figure 18 resembles the butterfly diagram displayed in
Figure 5. Most importantly, Spörer’s law of sunspot emergence at consecutively lower
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Jens Poppenborg
latitudes is in evidence. Furthermore, the different sunspot cycles vary in length as well
as in sunspot numbers, with very active cycles often being shorter than less active ones.
In addition, new sunspot cycles in reality as well as in the simulation usually start before
the last sunspots of the previous sunspot cycle have vanished.
One of the most striking differences is that in the observations, sunspots are often found
close to the equator by the time of sunspot maximum already. In contrast, during
the simulation they are still at latitudes of approximately 15◦ north and south of the
equator. Thus, the sunspot cycles in Figure 5 are usually spread over a wider range of
latitudes than during the simulation. This discrepancy is caused by the offset at which
new sunspots emerge that is migrating very slowly towards the equator. A more complex
set of rules for sunspot emergence could be used to solve this problem. For example,
new sunspots could not only emerge above the stored offset but also below.
Apart from this most obvious difference, I have numbered three features in Figure 18
where the simulation differs from the observations:
1. At this point, a single sunspot is slowly migrating toward the southern pole. Due
to a relatively slow decay rate, individual sunspots can remain on the sun’s surface
for a longer time than hitherto observed (cf. section 5.3.2). These spots can
occasionally either remain longer at one latitude than other sunspots or will make
a leap (e.g. migrating 2◦ within 2 months) towards higher latitudes. Nevertheless,
a similar behaviour of individual spots can also be seen in Figure 5, for example in
the years 1967 to 1968 where a single sunspot in the southern hemisphere migrates
towards the pole.
2. The 3rd sunspot cycle shows a very low activity in the southern hemisphere. While
this is consistent with some sunspot cycles in the butterfly diagram in Figure 5, in
the simulation it usually indicates that the magnetic field at the respective pole has
not been completely reversed in the previous cycle. This problem will be discussed
in more detail in section 5.2.
3. The last point marks a sunspot cycle where the offset at which sunspots appear in
the northern hemisphere has reached the equator, and new sunspots keep emerging
at the same latitudes creating a parallel to the equator up to about 15◦ . Here, too,
a similar behaviour can be observed in the sunspot cycles during the 1930s and
1940s.
Overall, while the features mentioned here are more distinct than they are in the butterfly
diagram in Figure 5, they can still be found in the observations. Therefore, the butterfly
diagram from the simulation is - despite less variation in sunspot numbers - still a
reasonable representation of reality. Nevertheless, further tinkering with the rules can
still improve the results, although at a certain point it is more tinkering than science.
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5. Results
University of Osnabrück
5.2. Magnetic Field Reversal
As described in section 4.1.1, the sun’s magnetic field is reversed every sunspot cycle
and thus determines the length of individual cycles. According to Babcock and Babcock
[1955], the total magnetic flux in each hemisphere is approximately Φ = 1014 Wb. The
value I have chosen, Φ = 9 · 1015 Wb, is almost 100 times as large. In Appendix A.1 the
effect of a value which still is 50 times as large as the amount of magnetic flux measured
by Babcock is displayed. Overall, these large values are required because the amount
of magnetic flux within sunspots is generally larger in the simulation than it would be
in real sunspots. Figure 19 displays the magnetic flux at the poles of both hemispheres
during the simulation.
16
1
x 10
0.8
Magnetic Flux in Wb
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
200
400
600
800 1000 1200
Time in Months
1400
1600
1800
Figure 19.: This image displays the magnetic flux at the poles of both hemispheres during
the simulation. The dotted line represents the northern, the solid line the
southern hemisphere. The horizontal lines show threshold values - the initial
value of the magnetic flux as well as the zero line where the solar maximum
is reached.
The image shows that the sun’s magnetic field is often not completely reversed when the
new sunspot cycle starts. While most of the time the reversed field reaches between 80%
to 100% of the initial field strength, for example the simulated 3rd sunspot cycle shows
a remarkably weak field in the southern hemisphere. This could also be noticed in the
resulting butterfly diagram (see Figure 18) where the sunspot activity during this cycle
was low as well. Overall, in about 30% of the sunspot cycles, the magnetic field of one
hemisphere amounts to only 50% of the initial value.
In addition, it can be seen that the zero-crossing (solar maximum) occurs during different
months for the northen and southern hemisphere. This is consistent with sunspot observations where there can be as much as one year difference between the solar maximum
in the southern and the northern hemisphere [Waldmeier, 1960].
In sum, while the variations at times are very large, the field is generally largely reversed.
A comparison with observed polar magnetic fields is not possible because no reliable
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5. Results
Jens Poppenborg
manetic field measurements at high solar latitudes are available because they would
require observers situated above the sun’s poles. The available high latitude observations
indicate that the field is variable and is not necessarily the same in both hemispheres
either.
5.3. Characteristics of Sunspots in the Simulation
The rules regarding the behaviour of sunspots during the simulation have been introduced in chapter 4. Here, I will visualise various characteristics resulting from these
rules.
5.3.1. Diameter of Sunspots
As described in section 2.1.4, the size and thus the diameter of sunspots is described by
a lognormal-distribution, of which only the rightmost downward slope is modeled by an
exponential function in the simulation (cf. section 4.1.2). In order to check whether the
diameter spectrum of the simulation can also be described by a lognormal-distribution,
I have used the Matlab-function fminsearch to calculate a fit. Figure 20 displays the
diameter spectrum of sunspots as well as the lognormal fit in a log-log graph. The lower
part of the fit could not be plotted as it consists of negative values.
1
∆(Number of Sunspots) / ∆(Diameter)
10
0
10
−1
10
−2
10
−3
10
0
10
1
10
Diameter in Cells
2
10
Figure 20.: Diameter spectrum for the sunspots of this simulation at their maximum
size (crosses) as well as a lognormal-distribution fit. The last part of the fit
could not be plotted as it included negative values.
The resulting fit curve in Figure 20 matches the values from the simulation quite well.
It is an essential problem in this simulation that the diameter of sunspots is discrete and
can never be less than 6000 km which equals one cell. This is twice the size of the smallest
sunspots that can be observed on the sun’s surface (cf. section 2.1). It has to be taken
into account as well that in reality, several smaller sunspots often form a group while in
the simulation only individual sunspots exist. Thus, the generally larger sunspots in the
simulation can be regarded as equivalent to sunspot groups in the observations.
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5. Results
University of Osnabrück
5.3.2. Lifetimes of Sunspots
In the simulation, sunspots can either dissipate or interact with sunspots of the opposite
polarity, both of which can result in the disappearance of a sunspot. While the first
process is based on a linear decay rate, the second is random. Figure 21 displays the
lifetimes of sunspots in months. During the simulation about 42% of the sunspots have
been on the sun’s surface for more than one month, and 14% for more than 5 months.
4
3
x 10
Number of Sunspots
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
Lifetime in Months
30
35
40
Figure 21.: This image displays the lifetimes of sunspots in months.
While the general trend of the curve can also be found for observed sunspot lifetimes,
the values themselves are very high. For example, as I have written in section 2.1.3,
more than 90% of all sunspot groups decay within less than one month. However, a
direct comparison of the results is not possible as in section 2.1.3 very small sunspots
are included as well. If only sunspots of a size similar to that used in the simulation
were included, the results would be more similar, albeit the lifetimes of sunspots in the
simulation would still be larger.
5.3.3. Latitude Drift of Sunspots
The latitude drift of sunspots was introduced in section 2.2.3. Sunspots at latitudes
higher than 20◦ north and south of the equator tend to migrate towards the poles while
sunspots at lower latitudes are more likely to drift towards the equator. Although in
the simulation the graph is slightly displaced towards higher latitudes, the general drift
motion as well as the velocity are approximately the same as can be seen in Figure 22.
The drift rate at latitudes of less than 5◦ as well as more than 45◦ are not reliable as
only a limited amount of sunspots appeared at these latitudes. On average, the drift
rate is either 0, 15◦ per month towards the equator or 0, 15◦ per month towards the poles
respectively.
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5. Results
Jens Poppenborg
60
Latitude in ° (N and S)
50
40
30
20
10
0
−0.4
−0.3
−0.2
−0.1
0
0.1
Latitude Drift in ° per Month
0.2
0.3
0.4
Figure 22.: The image displays the latitude drift of sunspots. The left side of the xaxis resembles a drift towards the equator while the right side of the x-axis
resembles a drift towards the poles.
5.3.4. Bipolar and Unipolar Sunspots
Figure 23 displays the percentage of bipolar and unipolar sunspots on the sun’s surface
during the simulation. On average, 64% of the sunspots are bipolar and 36% unipolar,
which is also consistent with the observations. It can be seen that at the beginning of
sunspots cycles there are more bipolar pairs while at the end of sunspot cycles, with the
last sunspots gradually dissipating, there are more unipolar sunspots.
100
90
Number of Sunspots in %
80
70
60
50
40
30
20
10
0
0
200
400
600
800 1000 1200
Time in Months
1400
1600
1800
Figure 23.: The percentage of bipolar (solid line) and unipolar (dotted line) sunspots on
the sun’s surface during the simulation.
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University of Osnabrück
5. Results
5.3.5. Discussion of Sunspot Characteristics
While the results for the latitude drift of sunspots as well as the proportion of unipolar
sunspots to bipolar sunspot pairs is consistent with the observations, the diameter of
sunspots as well as their lifetime are both larger than the observed values. The crucial
problem for these with parameters is the discretisation of time and space in the simulation. Thus, multiples of 6.000 km and one month for diameter and time respectively
cannot grasp the finer nuances.
Nevertheless, as the general trend in both cases has been modeled correctly, the parameters are satisfactory for the time of 160 years that is simulated here. Furthermore, the
main focus of this bachelor’s thesis is not on a perfect model of sunspots during the solar
cycle but rather on the accompanying effects such as the conversion of energy in solar
events.
Alternatively, a model simulating one or two sunspot cycles could be programmed which
uses a finer grid as well as a shorter time step (e.g. one day). As this would require
a revisal of all rules applied in the simulation, this will not be a part of this bachelor
thesis.
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5. Results
Jens Poppenborg
5.4. Sunspot Cycles in the Simulation
Last but not least, I will evaluate the sunspot cycles themselves. This includes a discussion of the sunspot numbers, the number of solar events, the energy conversion during
these events as well as the average cycle length based on the simulated results.
5.4.1. Number of Sunspots per Cycle
The number of sunspot groups during sunspot cycles varies between 2.000 and almost
5.000 groups per cycle [Li et al., 2002]. Figure 24 displays the number of sunspots in each
sunspot cycle of the simulation. This number varies between 2.500 and 4.000 sunspots,
which - although not as widespread as the number of sunspot groups in reality - is still
a good approximation. The number of sunspots during a sunspot cycle is limited by
the terminating conditions. As soon as the magnetic field of a hemisphere has been
reversed, no more sunspots will appear in that hemisphere. The variation in the sunspot
numbers is largely caused by magnetic flux which is released in solar events and thus
cannot be used to reverse the magnetic field. Also, cycles in which the magnetic field in
one hemisphere is not completely reversed show a lower sunspot number.
4000
3500
Number of Sunspots
3000
2500
2000
1500
1000
500
0
0
2
4
6
8
Sunspot Cycle
10
12
14
Figure 24.: This image displays the total number of sunspots during each sunspot cycle.
It should be noted that, while in reality sunspot groups and individual larger sunspots
are not the same, in this simulation I only use the latter. Thus, while the total number
of sunspots in reality can be far larger than 5.000, a lot of these sunspots are very small
and have a shorter lifetime than the sunspot group they are part of. Due to this, I will
regard the generally larger sunspots in the simulation as sunspot groups which in turn
are limited in their numbers. Furthermore, the large number of very small sunspots is
only has a relatively small significance regarding solar events and the reversal of the
sun’s magnetic field.
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5. Results
University of Osnabrück
5.4.2. Sunspot Numbers and Solar Events
During the simulation, the number of sunspots that could be observed on the sun’s
surface has been counted for every month as well as for every year. This sunspot number
is not the same as the number used in Figure 7, as in reality sunspot numbers are usually
processed using formulae such as the relative sunspot number described in section 2.2.4.
Furthermore, the released magnetic flux and the corresponding number of solar events
that occur when two sunspots of opposite polarity merge and neutralise magnetic flux
have been stored for each month as well as for each year.
Figure 25 displays the values that have been obtained on a monthly base while Figure
26 displays the corresponding values for each year.
Sunspots per Month
Sunspots
200
100
200
400
600
1600
1800
200
400
600
1600
1800
200
400
600
1600
1800
16
800 1000 1200 1400
Time in Months
Released Magnetic Flux per Month
10
14
10
12
10
Solar Events
Magnetic Flux in Wb
0
0
0
800 1000 1200 1400
Time in Months
Solar Events per Month
20
10
0
0
800 1000 1200
Time in Months
1400
Figure 25.: The top image displays the number of sunspots per month, the middle image
displays the released magnetic flux per month and the bottom image displays
the number of solar events per month.
Comparing the sunspot numbers from these images with those in Figure 7 it can be
seen that the general trend is the same. One difference, however, is that in reality, the
variations between the different cycles are greater than they are in the simulation. The
same applies to the length of the various cycles. Also, while in reality the number of
sunspots is smaller for longer periods of time it rises and falls faster in the simulation.
Apart from this, however, the variations between the cycles are present, even if not as
remarkable as in reality.
When regarding the released magnetic flux as well as the number of solar events in both
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5. Results
Jens Poppenborg
figures a periodicity similar to that of the sunspot cycle is obvious. A difference is that
the variation is far greater, and several peaks - for example large solar flares - are found
occasionally. Moreover, these peaks are often located near the solar minimum, at a
time when only few sunspots are on the sun’s surface. In the simulation, this is caused
by sunspots near the equator being more likely to merge with sunspots of the opposite
polarity across the equator (cf. section 4.2.2). The same can be observed in reality, where
solar flares and CMEs are occurring during the whole sunspot cycle. Figure 27 displays
the sunspot numbers (top image) as well as protons of E > 4 MeV (middle image) and
E > 60 MeV (bottom image) as they were measured by the Interplanetary Monitoring
Platforms (IMP) 2 and 6 (see http://nssdc.gsfc.nasa.gov/space/imp.html). The
raw data has been processed and visualised by Friedhelm Steinhilber. Here, too, it can
be seen that, while the general periodicity is still present, the maximums and minimums
do not necessarily coincide.
Sunspots per Year
Sunspots
1000
500
Solar Events
Magnetic Flux in Wb
0
0
20
40
20
40
20
40
16
60
80
100
120
Time in Years
Released Magnetic Flux per Year
140
160
120
140
160
120
140
160
10
15
10
14
10
0
60
80
100
Time in Years
Solar Events per Year
100
50
0
0
60
80
100
Time in Years
Figure 26.: The top image displays the number of sunspots per year, the middle image
displays the released magnetic flux per year and the bottom image displays
the number of solar events per year.
As for the number of solar events displayed in the bottom images, these are smaller
than the observed ones. Thus, the number is approximately as large as the number of
X-Flares 16 , but a lot smaller than the total count of solar flares and CMEs. This was to
be expected though, as the magnetic flux in this simulation vanishes immediately when
two sunspots of opposite polarity merge. In contrast to this, when two sunspots merge
in reality, the magnetic flux is neutralised in the course of several days and can result
16
These are very strong solar flares.
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5. Results
University of Osnabrück
in a larger number of solar events. Also, smaller events such as microflares cannot be
observed at all in this simulation as they do not necessarily require sunspots to form.
Figure 27.: The data displayed in this image has been recorded by Interplanetary Monitoring Platforms (IMP) and was processed and visualised by Friedhelm Steinhilber. The top image displays the total sunspot number, the middle image
displays the proton flux of protons with an energy of E > 4 MeV while
the bottom image displays the proton flux of protons with an energy of
E > 60 MeV.
Overall, the results displayed in this section are, albeit not always the same as in reality,
a very good approximation of the observations. Nevertheless, it still is possible to refine
the results and in particular the variations between the different sunspot cycles by further
adjustments of the rules.
5.4.3. Energy Conversion in Solar Events
During the simulation, whenever two sunspots of opposite polarity merge and release
magnetic flux, this flux is converted into energy. A part of this energy manifests itself
as solar flares and coronal mass ejections, while another part heats the surrounding
photosphere (cf. section 2.5). In order to calculate the released energy, I first calculate
the energy density (magnetic pressure):
pm =
B2
2 · µ0
(8)
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5. Results
Jens Poppenborg
The magnetic field B in this case is twice the magnetic field of the removed sunspot.
As described in section 4.1.2, the magnetic field and thus the magnetic flux of sunspots
is generally larger in this simulation than in reality as there is no distinction between
umbra and penumbra of sunspots. This will have a marked influence on the energy, as
the magnetic field is squared in Equation 8.
In the next step, I add the diameter d of the removed sunspot to the length l the other
sunspot shrinks in order to calculate the volume of the hemisphere above both sunspots.
This is based upon the very simplified assumption that solar flares and coronal mass
ejections form a hemisphere above the neutral line of the two reconnecting sunspots.
The volume of this hemisphere is
V =
2
· π · r3
3
(9)
with the radius r being (d+l) divided by two. Now, the released energy can be calculated
using the following equation:
E = pm · V
(10)
Figure 28 displays the energy that has been converted during the solar events of the
simulation.
4
1200
10
1000
3
800
Numbe of Events
Number of Events
10
600
2
10
400
1
10
200
0
0
0
5
Energy in J
10
27
x 10
10 20
10
25
10
Energie in J
30
10
Figure 28.: This histogram on the left side displays the energy that has been converted
during the simulation. Some events during which more energy has been
released are not displayed. On the right side, the same data is displayed in
a log-log graph.
It is obvious that the energy release during the events is generally larger than the 1025 J
observed during the largest solar flares in reality. The largest solar event during the
simulation converts an energy of 4 · 1030 J while the smallest converts 1023 J. Averagely,
44
5. Results
University of Osnabrück
3 · 1027 J are released in the solar events of the simulation. While this is significantly
larger than the observed energy release, the following factors have to be considered.
First of all, not all of the energy is released in solar events but some is also used for a
heating of the surrounding photosphere. For example, the photosphere around the flare
observed by Carrington in 1859 had to be considerably hotter than the photosphere at
approximately T = 5.800 K in order to be seen in white light. Secondly, as I have pointed
out earlier, the magnetic field of sunspots in the simulation is the same for both umbra as
well as penumbra. In reality, however, the magnetic field of the umbra can be as strong
as B = 3.000 G while the penumbra only has a magnetic field of B = 500 G. Finally,
the assumption that the volume of these solar events can be described by a hemisphere
is very simplified as well.
Considering these difficulties, the energy release during solar events in the simulation is,
despite being averagely 100 times as strong as the strongest solar flares in reality, still
sufficiently accurate. Also, smaller events such as microflares can not be detected in this
simulation at all.
5.4.4. Total Sunspot Area
Apart from the butterfly diagram, Figure 5 also features the daily average sunspot area
in percent of the visible hemisphere. Although a direct comparison is not possible, the
monthly total sunspot area in percent of the sun’s surface is displayed in Figure 29.
1.6
1.4
Sunspot Area in %
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800 1000 1200
Time in Months
1400
1600
1800
Figure 29.: The graph displays the monthly total sunspot area in percent of the sun’s
surface.
It can be seen that at no time during the simulation more than 2% of the sun’s surface
is covered by sunspots. While this is certainly more than the daily average sunspot
area displayed in Figure 5, a comparison with the monthly average sunspot area of the
full sun [Marshall Space Flight Center] shows that the values are still very small. For
example, during sunspot maximum about 8% to 15% of the sun’s surfcace can be covered
by sunspots.
45
5. Results
Jens Poppenborg
5.4.5. Average Cycle Length
Finally, in order to control the average length of the sunspot cycles, I used the function
fft 17 in Matlab in order to calculate the periodicity of the monthly sunspot data as
well as the monthly released magnetic flux. The results have been visualised in Figure
30 and Figure 31 for sunspot numbers and magnetic flux respectively. As can be seen
in both images, the periodicity is approximately 11 years and thus consistent with the
observations. Furthermore, the peaks are wide at their base, spanning between 9 and 14
years. This represents the variability of the cycle length in the simulation.
8
3.5
x 10
Period = 11.4226
3
Power
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
Period (Years/Cycle)
30
35
40
Figure 30.: This image displays the average length of sunspot cycles based on the
monthly sunspot numbers.
14
18
x 10
Period = 11.4226
16
14
Power
12
10
8
6
4
2
0
0
5
10
15
20
25
Period (Years/Cycle)
30
35
40
Figure 31.: This image displays the average length of sunspot cycle based on the monthly
released magnetic flux.
17
Fast Fourier Transformation.
46
6. Conclusion
There is a theory which states that if anybody
ever discovers exactly what the Universe is for
and why it is here, it will instantly disappear and
be replaced by something even more bizarre and
inexplicable.
There is another theory which states that this has
already happened.
Douglas Noel Adams
This bachelor thesis was concerned with the question whether it is possible to simulate
the solar cycle including the magnetic energy release based on a cellular automaton.
Although the term cellular automaton is used here in a rather liberal interpretation, the
above question can be answered positively: the butterfly diagram, a representation of
the solar cycle considering a rather large number of details, can be reproduced quite
reasonably. Most importantly, the energy conversion during solar events could be modeled sufficiently. Also, the periodicity of both, sunspot numbers as well as solar events
and released magnetic flux, is about eleven years for both despite a displacement of
minimums and maximum in the graphs. This, too, is consistent with the observations.
Nevertheless, differences such as the lifetime and diameter of sunspots, both of which
are larger in the simulation than in the observations, remain. This problem is caused
by the discretisation in time and space that are both handled in greater intervals than
could detect the finer nuances. Here, a finer lattice - for example with latitude strips
of 0.1◦ - as well as shorter time intervals might help. Apart from this the trend of the
resulting graphs for both, diameter as well as lifetime, is similar to those gained from
sunspot observations. Another detail that has been neglected in the simulation was, that
umbra and penumbra have a different magnetic flux density. In the simulation, however,
both have the same magnetic field. This has an influence on the sun’s magnetic field as
well as on the magnetic energy release, both of which are larger than they should be.
Therefore, a more accurate simulation should distinguish between umbra and penumbra
of sunspots.
In addition, the variations in sunspot numbers are more limited in the simulation than in
the observations. For example, in reality the number of sunspots as well as the number
of events are more intertwined. For example, during the very active solar maximum of
1958, not only the number of solar events but also the number of sunspots has been
very high. Although the same can be observed in the simulation, the peaks are less
remarkable than in reality. Here, better rules for the merging of two sunspots as they
could be implemented if a two-dimensional grid were used would improve the results.
However, a two-dimensional simulation would have to deal with several other problems
such as sympathetic flares and CMEs caused by disconnection events for which the cause
of their occurrence is not yet known.
47
6. Conclusion
Jens Poppenborg
Another difficulty with the simulation is its dependence upon randomness as well as the
abstraction of the rules. A more precise model of the solar cycle could be assembled using
a two-dimensional grid. This would allow for a more direct transfer of the observations
into the rules. But even this model would still be based on randomness. In addition, it
is questionable in how far such a model could - for example given the locations of the
first sunspots from the 24th sunspot cycle - predict the course of a sunspot cycle.
In conclusion it can be said that, while not perfect, the simulation is still capable of
reproducing various observations made about sunspots and the sunspot cycle. Overall,
it depends on the questions that are to be answered by the simulation which aspects
have to be refined. For this bachelor thesis, the goal was to simulate the magnetic energy
release and its temporal distribution on a time scale of centuries. The randomness that
accompanies the rules is of minor importance, as the reasons for the occurrence of solar
events are as of yet not fully understood, and thus could not even be modeled accurately.
In addition, since no observations of magnetic energy release exist for such long time
scales, the goal is not the reproduction or prediction of a certain solar cycle but rather
a case study: the simulation allows to produce a large number of different solar cycle
sequences. These can then be used in order to model corresponding reactions of the
earth’s atmosphere on the released energetic particles.
48
A. Parameters
Here, I will show the influences that various parameters have on the butterfly diagram
resulting from the simulation. While the results I visualise here are generally worse than
those I have evaluated in section 5, they still give an indication into which direction
a certain parameter can be tuned for better results of the simulation. However, one
should be aware that a lot more fine-tuning does not yield a physically more accurate
simulation because the simulation depends on a large number of parameter and transition
probabilities, none of which is known with high accuracy. As mentioned in the conclusion,
before one starts any fine-tuning one should clearly identify the question to be answered
with the simulation.
A.1. The Sun’s Magnetic Field
In section 4.1.1 I have written that a value is chosen which represents the magnetic field.
This value is, among other parameters, responsible for the length of sunspot cycles in
the simulation. Figure 32 displays the butterfly diagram if a smaller value is chosen,
Figure 33 for a larger one:
Figure 32.: The butterfly diagram if the sun’s magnetic field is half as large as it is in
the original simulation.
The influence of the magnetic field on the simulation is obvious in both pictures. While
smaller changes to the value go largely unnoticed, half the original value or twice the
original value visibly influence the length of the sunspot cycles. Also, with a larger
magnetic field, the chance that the magnetic field will not reach its initial value at the
end of a sunspot cycle increases.
49
A. Parameters
Jens Poppenborg
Figure 33.: The butterfly diagram if the sun’s magnetic field is twice as large as it is in
the original simulation.
A.2. Merging of Sunspots
In the original simulation, the chance that sunspots of different bipolar pairs and of
different polarity near the equator merge is about 90% while it is only 2% at higher
latitudes. Figure 34 displays the resulting butterfly diagram if the probabilities are only
70% near the equator and 15% at higher latitudes. This results in a smaller number of
sunspots per cycle, but also in a larger number of solar events.
Figure 34.: In this butterfly diagram, the probability that sunspots of opposite polarity
will merge at higher latitudes has been increased.
Overall, these parameters mostly affect the periodicty in the number of solar events. The
more similar both values are, the less distinction will be between the number of sunspots
and the release of magnetic flux during the simulation
50
University of Osnabrück
A. Parameters
A.3. Emergence of new Sunspots
A greater influence on the simulation has the probability with which new sunsots emerge
(cf. section 4.2.4). In Figure 35 I have chosen smaller probabilities of 10% before solar
maximum and 9◦ after solar maximum as well as larger probablities of 30% before solar
maximum and 25% after solar maximum for which the resulting butterfly diagram is
displayed in Figure 36.
Figure 35.: Butterfly diagram with reduced probabilities for sunspot emergence.
Figure 36.: Butterfly digram with increased probabilities for sunspot emergence.
Especially in Figure 36 the huge influence of the probability for the emergence of new
sunspots is visible. Already a minor change can cause enough sunspots to emerge and
reverse the sun’s magnetic field in half the time as before.
51
52
B. CD-ROM
On the CD-ROM that comes with this bachelor’s thesis the following can be found:
• The source code of the simulation can be found in the folder Code.
• The compiled program (*.class files) can be found in the folder Simulation. In order
to start the simulation, these files have to be copied to hard disk and the command
java Start has to be called. The results, apart from the butterfly diagram, will be
stored in text files in the same folder.
• This bachelor thesis as PDF-File in the folder Thesis.
• Java version 5.0 (J2SE) for Windows as well as a RPM in a self-extracting file for
Linux can be found in the folder Java.
53
54
C. Resources
C.1. Java
The simulation has been programmed using the Java 2 Platform Standard Edition 5.0
(J2SE 5.0) which includes the J2SE Development Kit 5.0. This version is only limited
downward compatible as it includes several new features that didn’t exist in earlier
versions. The latest version of Java can be downloaded at http://java.sun.com/. As
references I have used the book by Ullenboom [1988] as well as the J2SE API Specification
by Sun Microsystems, Inc..
C.2. Matlab
For the visualisation of data other than the butterfly diagram I have used Matlab 7.0
by The MathWorks. Matlab offers a wide variety of tools for numerical computing,
some of which have been helpful in evaluating data from the simulation.
C.3. LATEX
This bachelor thesis has been written using LaTeX as well as the KDE Integrated LaTeX
Environment (Kile).
C.4. Vim
Despite better development environments like Eclipse, I have written the entire simulation using the editor Vim which is available for free at http://www.vim.org/.
C.5. OpenOffice
Finally, the sketches in this bachelor thesis have been made using OpenOffice.org Draw.
OpenOffice.org is available for free at http://www.openoffice.org/.
55
56
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60
Acknowledgements
First of all, I would like to thank Prof. Dr. Kallenrode for her excellent support in
writing this bachelor’s thesis. She has had many a valuable comment to help me on my
way from the first idea to the finished documentation. Also, I have greatly enjoyed the
time I spend with the other members of the research group Numercial Physics: Modeling.
Secondly, I would like to thank my fellow students and friends who have accompanied
me through the past three years. Foremost, this includes Alexander Niemer, David
Engelhardt and Georg Hofmann without whom this time would never have been as
much fun as it was.
Last but not least, I would like to thank my parents who have supported me both
motivating as well as financial and still continue to do so.
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62
Eidesstattliche Erklärung
Hiermit versichere ich, die Bachelorarbeit selbstständig und lediglich unter Benutzung
der angegebenen Quellen und Hilfsmittel verfasst zu haben. Die Stellen der Arbeit, die
anderen Werken dem Wortlaut oder dem Sinn nach entnommen sind, habe ich unter
Angabe der Quellen der Entlehnung kenntlich gemacht. Dies gilt sinngemäß auch für
gelieferte Zeichnungen, Skizzen und bildliche Darstellungen und dergleichen.
Osnabrück, der 22. September 2006
63