Chapter 1
Introduction
The first literature to deal with the northern lights, that are one of the effects of
substorms, point as far back as to the times of the Old Testament. Since these
lights are visible with the naked eye (see Figure 1.1), discussion started early to
what caused this effect. It appears that Galileo Galileo coined the phrase aurora
borealis, meaning ”Dawn of the North”. In 1731, the French philosopher de
Mairan suggested that the aurora was connected to sunspots. After that time,
studies of geomagnetism and the aurora became more firmly linked.
With the invention of the compass the study of geomagnetism started and
knowledge about the Earth’s magnetic field and its details such as declination
and inclination increased. It was though not until after the telescope was invented
that it was possible to collect information about the sun. After the Maunder
Minimum (about 1645 to 1700), during which there were barely any sunspots
visible, the study of the sun could really start. In 1741 O. Hiorter discovered that
geomagnetic and auroral activities were correlated. In 1860 Elias Loomis of Yale
noted that the auroral zone is an oval band around the magnetic pole, roughly 20
- 25 from the pole. But only in 1897 was discovered by Kristian Birkeland (after
whom the cross-tail current or Birkeland current is named), that large electric
currents flow along magnetic field lines during the aurora.
In the following years more details about the aurora and ionosphere were worked
out until interest in the upper extension of the ionosphere grew, the area today
known as the magnetosphere. Around 1962 the idea of a plasma of electrons
and ions coming from the sun was established and soon after that, at the dawn of
the space age, space probes were able to confirm the theories. (For more
Information, please refer to [Introduction to Space Physics]).
In 1964 Akasofu wrote one of the first papers about auroral substorms [Akasofu
1964]. Considering the history of the aurora and that it was linked to the sun, it is
not surprising that it was originally thought to be part of a sun storm. Over the
years as the knowledge about substorms increased, it became clear though that
a substorm can occur as part of a storm, but does not have to be linked to it.
This introductory chapter will in more depth describe the details of substorms,
how they occur, and what their effects are.
1.1 The solar wind and the magnetosphere
The Sun releases a flow of ionized solar plasma and a remnant of the solar
magnetic field, that is called the solar wind. It flows radially outward from the Sun
at 1 AU. The solar wind typically has a velocity of around 400 km/s, a number
density of 5 #/cc, a temperature of 64,000 K, and a magnetic field strength of 5
nT. The underlying magnetic pattern of the sun rotates every 27 days. The
complex field structure of the sun’s photosphere simplifies with increasing height
in the corona until a single line separates the magnetic field that points toward or
away from the sun. Figure 1.2 shows an artists depiction of this so-called
heliospheric current sheet that separates regions of solar wind with the two
different polarities [WSO].
This picture of the solar wind can nicely explain the idea that the solar wind can
have differing directional components. We can see how Earth (depicted as the
sphere in the right corner of Figure 1.2) switches from a region with one type of
polarity to a region with the opposite polarity, as it circles the sun. Just as we
know it from wind on Earth, the solar wind can reach our planet with a southward
or northward component. Before it can actually hit Earth though, the solar wind
interacts with the planet’s magnetic field. Our planet’s magnetic field could be
approximated by a dipole field with an axis tilted by about 11 degrees from the
rotation axis, if there was no solar wind. One can imagine the Earth with its
magnetic field to lie within the flow of the solar wind, just like a stone in a stream
of water. The solar wind compresses the Earth’s magnetic field on the side that is
closer to the sun (day side) and drags it out into a long tail on the side that is
turned away from the sun (night side). This scenario can be seen in Figure 1.3.
As the supersonic solar wind reaches Earth its dynamic pressure, or momentum
flux, compresses on the Earth’s magnetic field, confining it on its day side and
stretching it into a long tail on its night side. Earth’s magnetic field pressure
establishes an equilibrium with the Sun’s plasma pressure, so that the size of the
day side magnetosphere depends on the strength of the solar wind flux.
Magnetosphere is the term for the region of space that is dominated by Earth’s
magnetic field. The equilibrium between the solar wind and Earth’s magnetic field
can be described as follows:
ρSW u2SW = (B2MS) / (2 μ0)
where the subscripts SW and MS refer to the solar wind and the magnetosphere,
respectively, and uSW is the solar wind’s group velocity.
Because the solar wind is supersonic, it creates a shock front, known as the bow
shock in front of Earth as depicted in Figure 1.3, that slows down the solar wind.
The interaction of the solar wind with Earth’s magnetic field is complicated and its
discussion exceed the scope of this dissertation. The basic result is that Earth’s
magnetic field shields the planet from the particles from the sun, that can only
enter into our ionosphere through the poles, an area that is called polar cusp.
The magnetic field around Earth separates into different regions, depending on
their properties. The magnetopause is the boundary between solar wind field
lines and field lines that are connected to Earth. If only one end of a fieldlines is
connected to Earth, the field line is called open, if both ends are connected the
field line is considered closed.
The area between the bow shock and the magnetopause is called
magnetosheath. All its field lines are solar wind field lines and are not connected
to Earth. The velocities in this region are higher than in the magnetosphere. The
two main regions of interest for this thesis are on the night side of the
magnetosphere - the plasma sheet and the lobes. The part of the
magnetosphere in which they lie is called the magnetotail. It extends from Earth
in the opposite direction of the Sun at least three times the distance from the
Earth to the moon [Lopez 1990].
The high density plasma sheet is the area of entirely closed field lines. Its
magnetic field has a high Bz and low Bx component, due to the shape of the field
lines. The amount of thermal energy Eth = nkT in this region is about as high as
the amount of magnetic energy Emgn = B2/2μ0. The amount of kinetic energy Ekin
= ½ρv2 is less than both. The β-factor, which is the ration of thermal over
magnetic energy is around 0.5.
The lobe regions consists primarily of magnetic energy with the magnetic field
consisting of a high Bx component and low Bz component. The magnetic field in
the northern lobe points towards Earth (Bx > 0), whereas the magnetic field in the
southern lobe points away from Earth (Bx < 0). The field lines in the lobe are all
open. The β -factor of the lobes is very low (< 0.1) since the magnetic energy in
this region greatly outweighs the amount of both thermal and kinetic energies.
1.2 What is a substorm?
When the solar wind magnetic field has a southward component (Bz < 0), it can
connect to the Earth’s northward pointing magnetic field. This process is called
reconnection. It was first proposed by Dungey in 1961. Reconnection is a
process by which magnetic energy is converted into plasma energy. The frozen-
in-flux criterion breaks down, allowing the solar wind field lines and Earth field
lines to enter the diffusion region from right and left, and instead of being
annihilated, leave from both sides. The field lines are being ”cut” and
reconnected” to different partners. Plasma originally on different flux tubes now
find itself on a single flux tube. This process is shown in Figure 1.4.
The oppositely directed magnetic field lines of the IMF and Earth can connect at
the subsolar point to each other and open solar wind field lines can become
connected to Earth. The place where this interconnection occurs is called an X-
line because of its magnetic geometry. The newly merged field lines convect over
the northern (southern) pole to the night side. This process transfers magnetic
flux into the magnetotail, which is equivalent to storing energy in the magnetotail
since the energy density (in J/m3) of a given amount of magnetic flux density is
B2/2μ0, where B is the magnetic flux density (in nT/m2) and μ0 is the permeability
of free space. During this time the polar cap increases. The polar cap is the area
in the poles in which all field lines are open. The polar cap flux describes the
amount of flux that comes in or goes out of the polar cap. As reconnection occurs
on the day side of Earth, IMF field lines and closed field lines reconnect, creating
additional open field lines. The more open field lines the larger the polar cap, and
therefore the larger the polar cap flux.
The polar cap flux is defined as
Φm = Int (B · ds)
where the Earth’s magnetic field is
B(R,Θ) = −B0R3sin(Θ) eΘ − 2B0R3cosΘ) eΦ
with r[m] = R[RE] and the area element is
ds = r2sinΘ dΘ dΦ er
The result of this integral leads to
Φm = Int (B · ds) = (-2B0r2)/(R3) Int [dΦ’] Int [cos(Θ’)sin(Θ ‘)dΘ’]
= Σ B0(RE)2(sin2ΘΔΦ)n
B0 is a constant of the value 3.1·10-5 T and RE is the Earth Radius of 6378 km, Θ
is the co-latitude and Φ as the radial angle. Using simulated magnetometer
stations distributed every 10 (ΔΦ = 10) around the polar cap at a specific co-
latitude Θ the polar cap can be precisely determined.
The energy that has been added to the magnetotail due to the day side
reconnection process has to be released eventually. This process is what is
called a substorm (e.g., [Lopez 1990]).
A substorm has three phases:
* Growth Phase
* Expansion Phase
* Recovery Phase
The growth phase is the phase during which energy is being added to the
magnetotail, as just described. It starts with a southward turning of the IMF and
ends at substorm onset. During the growth phase open field lines are being
created by the reconnection process at the day side of Earth and are then
convected over the poles to the tail, eroding the day side magnetosphere of
magnetic flux. In the tail the amount of open field lines and therefore the
magnetic field in the lobes increases, the magnetotail becomes very stretched
while the plasma sheet thins and is being compressed. During this period the
cross tail currents strengthens. The cross tail current is a current that flows
across the magnetotail due to a pressure gradient between the lobes.
The Substorm Onset is the point when the stored energy is being released and is
therefore the start of the expansion phase. How and where this release starts is
still under discussion [Nagai et al., 2005] and I will for now just describe the
observations that are being associated with the expansion phase. The two main
theories that describe the physics of what happens during the expansion phase
will be explained in more detail in section 1.3.
There are four main observations that are associated with a substorm expansion
phase. They are not presented in any specific order.
The first observation is the brightening and expansion of the pre-existing auroral
arc [Akasofu 1964]. The auroral arc, that is visible with the naked eye, will
expand poleward, producing a pattern that is similar for every substorm.
The second observation is the dipolarization of Earth’s magnetic field.
Dipolarization means that the magnetic field returns to a more dipolar state than
the elongated shape that it had during the growth phase. This effect can be
measured with any satellite that is in the right position at the right time.
The third effect is the generation of a new current system [McPherron et al.,
1973], that is called the substorm current wedge. This scenario is depicted in
Figure 1.5 (taken from [Lopez 1990]). One can see in pink the cross tail currents.
In blue are the field aligned currents that flow along the magnetic field lines. As
the substorm current wedge starts, the during the growth phase enhanced cross
tail currents, are being diverted via the field aligned currents into the ionosphere.
To close the current system, the current flows westward through the ionosphere
(depicted in green). It leaves the ionosphere again via the field aligned currents
to join the cross tail currents in the magnetotail. The part of the current that flows
in the ionosphere is called the westward electrojet and is measurable by ground
based magnetometer stations, that are underneath the current, via positive or
negative bays in the horizontal (H)- and eastward (D)-components of the
magnetic field. The onset and decay of the substorm current wedge is also
measurable by magnetometers on satellites, as long as the satellites are properly
positioned in the magnetotail.
Observation number four is the release of a plasmoid down the tail. A plasmoid is
a disconnected tangle of magnetic field lines that is initially enclosed within the
closed field line region, when reconnection is purely on closed field lines. Figure
1.6 (taken from [Lopez 1990]) shows a drawing of this process. Due to the very
thin plasma sheet it is possible that the two sides of a closed field line come so
close to each other that they build an X-line and reconnect to each other. This
way they create an area of closed field lines that are not connected to Earth (an
O-line), but only to themselves. As reconnection proceeds to open field lines, the
plasmoid is ejected down the tail by both magnetic tension and flow pressure
from the reconnection region.
Other observations that have been made during the substorm expansion phase
are that it goes along with a reduction of the polar cap flux, speaking for
reconnection in the magnetotail. Also the acceleration of particles, sometimes up
to several MeV [Lopez and Baker, 1994] has been observed. It seems that the
onset of a substorm is usually triggered, either by a pressure pulse that hits the
day side magnetosphere or by a northward turning of the solar wind.
The last phase of a substorm is the expansion phase. As all the energy has been
released, lobe reconnection stops, the substorm current wedge decays, and the
auroral bulge relaxes again. In this phase the magnetic field and all other
properties of the magnetotail return back to the original values that they had
before the start of the growth phase.
1.3 Substorm Models
There are two competing models that try to describe the observations during a
substorm. The first one is the ”Near Earth Neutral Line (NENL) Model”. This
model was first proposed by Hones [1979] but has undergone tremendous
changes since then. The most recent version was written by Baker et al. [1996]
and is the one that will be explained here. The second model is the ”Current
Disruption (CD) Model” that was proposed by Lui [1991].
For both models growth phase and recovery phase have basically the same
properties. The growth phase starts with the southward turning of the
Interplanetary Magnetic Field and leads to day side reconnection that loads
energy into the magnetosphere, as described in section 1.2. The main difference
is that the CD model focuses more on the changes in the near-Earth region. It
assumes that the cross tail current in the near-Earth region will increase more
than in the mid-tail, due to this being the area in which the plasma sheet
undergoes the largest amount of change. This idea directly links the field aligned
currents associated with the region of high current density into the pre-existing
auroral arc that brightens at substorm onset. In the NENL model it is the added
flux to the polar cap that allows the polar cap potential to increase and the aurora
to brighten [Wiltberger 1998].
The recovery phase in both models is the phase during which the near-Earth
neutral line moves tailward and the plasma sheet thickens. Signatures of a
plasmoid that retreats tailward are measurable in the mid- to distant-tail regions.
It takes around two hours for the system to return to its quiet stage after a
substorm. The growth phase is still one of the least studied parts of a substorm,
primarily due to a limited amount of data available for this time [Wiltberger 1998].
The Near-Earth Neutral Line Model
The NENL model is the first comprehensive model for substorms and still the
most widely accepted. It assumes that it is lobe reconnection that starts the onset
of a substorm. When the plasma sheet has thinned sufficiently spontaneous
reconnection will occur in the magnetotail on closed field lines. The reconnection
starts slowly, since the rate of reconnection is inversely proportional to the
plasma density [Petschek 1966]. As reconnection moves onto open field lines,
with lower plasma density, the rate of reconnection will increase dramatically.
The newly formed X-line disrupts the cross-tail current and forms a plasmoid.
The formation of the reconnection region causes a reduction in the current
carrying capacity of the region. To maintain continuity the cross tail current has to
be diverted. This necessary diversion together with the velocity shears along the
edges of the flow jets, is what then creates the substorm current wedge
[Wiltberger 1998]. The reconnected field lines earthward of the neutral line snap
back back to Earth, dipolarizing the field and expanding the plasma sheet.
Tailward of the neutral line the plasma sheet thins rapidly and moves the
plasmoid down the tail. The near-Earth neutral line now becomes the distant
neutral line and the plasma sheet continues to expand.
The open field line reconnection in the mid-tail region in the NENL Model is what
releases the energy that has been stored in the magnetotail during the growth
phase. It creates the substorm current wedge, leads to the auroral brightening,
and creates the plasmoid.
The Current Disruption Model
In the CD Model the onset of the substorm starts with a disruption of the intense
cross tail current in the near-Earth region. Only if the ionosphere has a favorable
condition and a thin current sheet has developed in the tail, can the expansion
phase start. Otherwise a pseudo-breakup will occur and the expansion will not
proceed. As the expansion phase continues, multiple current disruptions occur,
that explain the intensifications and breakups on auroral arcs poleward of the
original breakup arc.
The current disruption is what forces most of the cross-tail current to be diverted
into the ionosphere. It will flow preferentially along pre-existing auroral arcs
because they provide better conductivity channels. The created substorm current
wedge causes a relaxation of the stretched magnetic field. This produces an
earthward convection surge that dipolarizes the Earth’s magnetic field. The
earthward transport of plasma is associated with movement of flux tubes. This
earthward surge of plasma will create a region tailward of the current disruption
region, that is partially evacuated. This will create a rarefaction wave that
propagates down the magnetotail and produces the observed thinning of the
plasma sheet in the mid-tail region. Earthward of the reconnection site the
plasma sheet begins to thicken while one or more plasmoids are formed and
ejected downstream. The substorm recovery phase starts when the plasma
sheet becomes thick enough to hinder reconnection to continue.
In the CD Model it is the current disruption that leads to substorm onset. It
creates the substorm current wedge, leads to auroral brightening and
reconnection, and dipolarizes the magnetic field.
The main differences between the NENL and CD models are that in the NENL
model the substorm onset starts with reconnection. All the other features are
being caused by this process. In the CD model onset starts with a disruption of
the strongly enhanced cross-tail current in the near-Earth region. The other
features such as plasmoids and reconnection are only part of the later
development of the expansion phase when the plasma sheet is thinned
substantially in the mid-tail region by rarefaction waves.
Problems with the Models
Both the CD and the NENL models have their problems. Where the NENL model
has problems with the timing of the auroral intensifications vs. the start of
reconnection and explaining some of the near-Earth phenomena, the CD model
has problems with its energy considerations.
Observations indicate that there is a delay between the brightening of the auroral
arc and the movement of the polar cap boundary. If open field line reconnection
is what causes auroral brightening in the NENL model, then this brightening
should occur simultaneously with the movement of the polar cap boundary. This
can be partially explained by mapping the active region of the ionosphere to the
near-Earth tail that is influenced by neutral line flows [Wiltberger 1998]. One of
the most difficult parts to explain is that the equatorward most discrete arc
brightens at onset. In terms of the NENL model this would mean that the
earthward directed flow jets pass through regions in the magnetosphere, where
the poleward arcs map, without any effect until they reach the equatorward arc.
There are definitely some more questions that need to be clarified to improve the
NENL Model. In general though, it explains most of the physical findings and a
wide range of observations fit naturally into the NENL framework (e.g.
[Angelopoulos et al., 1994,Baker et al., 1997]).
The CD Model has its biggest problems with its energy considerations. For the
current disruption to be able to cause the substorm onset, the near-Earth tail has
to be a sufficient source of energy for the substorm expansion phase. Hesse and
Birn [1993] have shown though, that there is not enough energy in the closed
field line region of the inner plasma sheet to power a substorm expansion. Also
the CD model does not take the polar cap observations properly into account,
that indicate that open flux is being converted into closed flux before the recovery
phase. It is difficult to understand in this model how the return of the lobe field to
pre-substorm values happens during the expansion phase if there is no open flux
being reconnected during this phase.
1.4 Substorm Multipoint Observations
What exactly causes the onset of the substorm, how and when the substorm
current wedge is generated, and where the reconnection region is exactly located
are still under discussion [Nagai et al., 2005]. One of the difficulties in resolving
these issues is that rarely are there sufficient spacecraft in the right position at
the right time to be able to determine the proper sequence of events. Moreover is
it difficult to determine the global evolution of the magnetosphere during a
substorm from single-point measurements.
To determine the evolution of a substorm multipoint observations are necessary.
Sometimes it is possible to find up to four satellites distributed in the tail that can
measure the evolution and development of the substorm current wedge.
Together with ground based magnetometer data it is then possible to analyze the
global effects of a substorm on the plasma sheet and the lobes and their timing.
The amount of good substorms with single-point observations though greatly
outweighs the ones with multi-point observations. Some of the satellites that are
good for substorm analysis are the GOES satellites, that are in geosynchronous
orbit, one over the east and one over the west coast of the United States. Due to
their location and orbit they usually see similar signatures of the substorm current
wedge as the mid-latitude stations. Satellites that have orbits around Earth and
can sometimes be found in the regions of interest are Geotail, Interball, and IMP-
8. They can measure e.g. high speed particle flows and changes in the magnetic
field and their data is easily available via the internet through CDAWeb.
Especially measurements of these satellites in the lobes have previously been
used to estimate the timing of the substorm onset. Previous publications [Baker
et al., 1985,Baker et al., 1997] have assumed that at substorm onset the lobe
magnetic field decreases and have used this point of field reduction as a
measure for onset timing. This is not always the case, as will be shown in
Chapter 3. Details about the satellites used in this dissertation can be found in
Appendix C.
There are of course more satellites available than the ones that were used for
this analysis. Nonetheless is it very difficult to find a large number of satellites in
interesting positions during the brief period of a good substorm. Some of the
papers that have done multi-point analyses of substorms are from Lopez, Lui,
and Ohtani. Lopez and Lui [1990] studied observations made by four spacecrafts
and two ground stations. They found that the disruption of the cross-tail current
sheet, the formation of the substorm current wedge, and the expansion of the
plasma sheet began in the near-Earth region, and subsequently spread tailward
as well as longitudinally. One can also determine the difference between a
pseudo-substorm onset and a standard onset, as Ohtani et al. [1993] have
shown in their paper, for which they found 6 satellites including the two GOES
that measured a current disruption that did not expand radially and decreased in
magnitude ad it expanded. They concur thus that the major difference between a
pseudo-substorm and a substorm is the absence of the expansion of the current
disruption, rather than the scale of the onset region. It is also possible to
measure properties of the westward traveling surge and to connect the activity in
the magnetotail to the surge pattern in the aurora [Lopez et al., 1990]. But even
though the current disruption region can nicely be measured by multipoint
observation, it is still an open question whether the formation of the reconnection
region precedes or follows the disruption of the cross tail current sheet.
Multipoint Observations are nevertheless only snapshots. Satellites can give an
excellent view on the micro scale level, but even with several satellites is it still
difficult to capture dynamics in the magnetotail on larger scales, e.g. the energy
dynamics and distribution during substorms. Originally it was thought that the
energy loading and unloading processes were different during the growth and
expansion phase of the substorm [Baker et al., 1985], but then it was clarified
that both are due to reconnection processes of open and closed field lines in the
day- and night side of Earth, respectively [Baker et al., 1996].
An interesting mission that has been launched in February of 2007 is the
THEMIS Mission [THEMIS]. A group of five identically equipped satellites will
orbit around Earth to get a look specifically at the effects of substorms on the
day- and night side magnetosphere. In the time from December until April will the
satellites be aligned like on a string of pearls down the magnetotail on the night
side of Earth with the goal to establish when and where substorms begin,
determine how substorms power the aurora, and identify how local current
disruption mechanisms couple to the more global substorm phenomena. The rest
of the year will the satellites be on other sides of Earth to conduct day side
science and radiation belt science. This project has the prospect to greatly
enhance the knowledge about substorms and hopefully clarify some of the
questions that are currently being under so much discussion. Unfortunately due
to its launch date it is not possible to include THEMIS substorms into this
dissertation, although this data would have nicely contributed to the results.
Due to the fact that even multipoint observations can only capture small pieces of
what happens during substorms, and can especially not say much about the
global energy distribution, this dissertation uses simulated substorms for its
results. The simulation that can give insight into the macro scale effects of
substorms together with satellite data for the micro scale effects, can give a
global detailed picture of what happened during the three phases of the substorm
and in what timely order the different effects happened.
1.5 Energy Calculations
Calculating the energy in the magnetosphere is tricky. There are no direct
observational means to determine the energy transfer from the solar wind to the
magnetosphere. Usually estimates of the energy budget of a substorm [Baker et
al., 1985,Tanskanen et al., 2002] are made via single point measurements in
space, estimates of the polar cap area based on space-based images, and
empirical relationships that relate quantities such as ground-based
magnetometer observations to energy dissipation. The first person to try and
estimate the energy transfer was Akasofu. The search lead to the epsilon (ε)
parameter (e.g. [Akasofu 1996]). The ε parameter is generally understood to
describe the transfer of solar wind Poynting flux into the magnetosphere
[Koskinen and Tanskanen, 2002]. The ε parameter is often given in the following
form in SI units:
ε = (4π/μ0) v B2 sin4(Θ/2)l02 (1.1)
where v is the solar wind speed in km/s and B is the magnetic field in nT. Θ is the
clock angle of the IMF orientation perpendicular to the Sun-Earth line, i.e. tan Θ =
By/Bz. l0 is an empirical scaling factor with units of length. But the factor l0 in ε is
very uncertain [Koskinen and Tanskanen, 2002]. It was introduced so that ε could
have units of energy and is usually taken to be around 7 RE. Akasofu [1981]
states that a level of input power greater than 100 GW (10 11 W) is substorm level,
meaning that at these levels substorms are likely to occur. Perreault and Akasofu
[1978] have originally introduced the ε parameter. They expressed the IMF
through the Poynting Flux and found a relationship to the energy consumption in
the magnetosphere:
ε = ( |E| |B| / 4π ) sin4(Θ/2) l02 (1.2)
This expression is neither in SI nor in cgs units, in which the Poynting vector S =
(c/4π)E × B, or S = E × B/μ0, respectively, but appears to be in the emu or esu
system [Koskinen and Tanskanen, 2002]. In these units one can transform the
electric field in equation 1.2 by using E = vB to
ε = (1/4π) v B2 sin4(Θ/2)l02
which is only a factor 4π smaller than the widely used expression 1.1. It is
possible that the factor 4π was interpreted to be hidden in the factor l0 [Koskinen
and Tanskanen, 2002].
That is not the only problem with the ε parameter, though. Besides that l0 is an
empirical parameter that is in the equation for pure unit reasons, the Poynting
vector is always perpendicular to the magnetic field. This can lead to a zero
Poynting flux in extreme cases of the solar wind, and then the ε parameter has to
be treated slightly different [Koskinen and Tanskanen, 2002]. Also equation 1.1
shows that the energy transfer from the solar wind to the magnetosphere does
not depend on the solar wind density. It only involves physical conversion of solar
wind kinetic energy to magnetic energy measured inside the magnetopause. This
is not the case though in reality. Lopez et al. [2004] and Palmroth et al. [2004]
have shown that the energy transfer does depend on the solar wind density and
cannot just be excluded.
Comparing the ε parameter to global MHD simulations has been done by
Palmroth et al. [2003] and Pulkkinen et al. [2006]. Palmroth et al. [2003] have
shown that the " parameter can sometimes match the simulation results nicely as
e.g. during the main phase of a storm [Palmroth et al., 2003], but for other
simulations it is not as good. For example during the late phases of a substorm it
gives a very poor estimate of the energy input [Pulkkinen et al., 2006].
Summing up, it is very clear that the ε parameter should be handled with care.
What is needed is a way to capture the global behavior of the transfer of energy,
in this case specifically during a substorm. To achieve this global picture of the
energy distribution and dynamics in the magnetosphere, this dissertation will use
the Lyon-Fedder-Mobarry magneto-hydrodynamic code to simulate three
substorms and calculate the energy in the plasma sheet and lobes during these
events. The analysis includes calculations of the polar cap flux, lobe energy and
plasma sheet energy during growth and expansion phase of the substorms. The
next chapter will go into the details of the simulation and will explain how plasma
sheet and lobes were chosen and the process how their total energy was
calculated.