3D Magnetic Nulls and Regions of
Strong Current in the Earth’s
Magnetosphere
by
Elin Eriksson
6 May, 2016
DEPARTMENT OF PHYSICS AND ASTRONOMY
UPPSALA UNIVERSITY
SE-75120 UPPSALA, SWEDEN
Submitted to the Faculty of Science and Technology, Uppsala University
in partial fulfillment of the requirements for the degree of
Licentiate of Philosophy in Physics
c
Elin Eriksson, 2016
Abstract
Plasma, a gas of charged particles exhibiting collective behaviour, can be
found everywhere in our vast Universe. The characteristics of plasma in very
distant parts of the Universe can be similar to characteristics in our solar
system and near-Earth space. We can therefore gain an understanding of
what happens in astrophysical plasmas by studying processes occurring in
near Earth space, an environment much easier to reach.
Large volumes in space are filled with plasma and when di↵erent plasmas
interact distinct boundaries are often created. Many important physical processes, for example particle acceleration, occur at these boundaries. Thus,
it is very important to study and understand such boundaries. In Paper I
we study magnetic nulls, regions of vanishing magnetic fields, that form inside boundaries separating plasmas with di↵erent magnetic field orientations.
For the first time, a statistical study of magnetic nulls in the Earth’s nightside magnetosphere has been done by using simultaneous measurements from
all four Cluster spacecraft. We find that magnetic nulls occur both in the
magnetopause and the magnetotail. In addition, we introduce a method to
determine the reliability of the type identification of the observed nulls. In
the manuscript of Paper II we study a di↵erent boundary, the shocked solar
wind plasma in the magnetosheath, using the new Magnetospheric Multiscale
mission. We show that a region of strong current in the form of a current sheet
is forming inside the turbulent magnetosheath behind a quasi-parallel shock.
The strong current sheet can be related to the jets with extreme dynamic
pressure, several times that of the undisturbed solar wind dynamic pressure.
The current sheet is also associated with electron acceleration parallel to the
background magnetic field. In addition, the current sheet satisfies the Walén
relation suggesting that plasmas on both sides of the current region are magnetically connected. We speculate on the formation mechanisms of the current
sheet and the physical processes inside and around the current sheet.
keywords: Magnetic Nulls, Particle Acceleration, Current sheets, Magnetic
reconnection, Magnetosphere.
List of papers
This thesis is based on the following papers, which are referred to in the text
by their roman numerals. Paper I is published, while Paper II is a manuscript
to be submitted.
I E. Eriksson, A. Vaivads, Yu. V. Khotyaintsev, V. M. Khotyayintsev,
and M. André (2015), Statistics and accuracy of magnetic null
identification in multi-spacecraft data, Geophys. Res. Lett.,42,
6883-6889
II
E. Eriksson, A. Vaivads, D. B. Graham, Y.V. Khotyaintsev, E.
Yordanova, M. André, C. Russell, R. Torbert, B. Giles, C. Pollock,
P-A. Lindqvist, R. Ergun, W. Magnes, and J. Burch (2016), Kinetic
Study of Thin Current Sheet in Magnetosheath Jet, Manuscript
Reprints were made with permission from the publisher.
Papers not included in the thesis
1. H. S. Fu, J. B. Cao, A. Vaivads, Y. V. Khotyaintsev, M. André, M. Dunlop,
W. L. Liu, H. Y. Lu, S. Y. Huang, Y. D. Ma, and E. Eriksson (2016), Identifying magnetic reconnection events using the FOTE method, J. Geophys.
Res. Space Physics, 121
2. V. Olshevsky, A. Divin, E. Eriksson, S. Markidis, and G. Lapenta (2015),
Energy Dissipation in Magnetic Null Points At Kinetic Scales, Astrophysical Journal, 807, 155
Contents
1
Introduction
..................................................................................................
1
2
Basic plasma physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Electromagnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
3
5
3
Magnetosphere
6
4
Magnetic Reconnection
5
The Cluster mission
6
The Magnetospheric Multiscale mission
.................................................
11
7
Magnetic Nulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Description of Magnetic Nulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Different Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
Location Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1
Poincaré Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2
Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Statistical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5
Reliability Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13
13
15
16
17
18
18
8
Plasma Heating and Particle Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Basics Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1
Betatron Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2
Fermi Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.3
Wave-particle Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Current Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
21
22
23
25
25
9
Looking Forward
26
.............................................................................................
...................................................................................
.......................................................................................
10 Acknowledgments
References
...............................................................................
8
10
.....................................................................................
27
........................................................................................................
28
1. Introduction
Space has always been (and still is) a very fascinating subject. Even with
the advantages in technology making it possible to make more and more detailed studies into what is actually happening, there is still many unanswered
questions. This thesis is a small contribution toward answering some of those
questions.
Plasma, a gas of charged particles exhibiting collective behaviour, exists
almost everywhere in our vast Universe. On Earth, the most common plasma
phenomenon is lighting [31]. In the late 1950 and 1960s in situ observations
of space plasma became possible with the advances in rocket and spacecraft
technology [37]. Despite the fact that near space and astrophysical plasmas
show very different ranges in their magnetic fields and plasma densities, unitless parameters characterizing those plasma environments, such as the plasma
beta, can be very similar to each other [57]. Thus, studying processes occurring near Earth, a much easier to reach environment, can help us understand
what occurs at other places in the Universe.
Understanding near space is not easy, so it should not come as a surprise
that it is a collaborative effort. Both observational (laboratory and space) and
simulation communities need to work together to understand what is happening in space. Simulations results are commonly used by comparing them to
space observations. Powerful computer simulations can test out different theoretical models and conditions, which is helpful when looking at processes
occurring in large systems where the basic theory is too complicated to use.
In this thesis, we present a multi-spacecraft study of magnetic nulls, regions
of vanishing magnetic field, in the Earth’s nightside magnetosphere (Paper I)
and a kinetic study of a thin region of strong current in the turbulent magnetosheath and its related electron acceleration (Paper II). Both structures are
essential in two-dimensional (2D) reconnection and are also believed to be
important in three-dimensional (3D) reconnection [49].
In the following chapters, we begin by giving a basic introduction to plasma
physics and the Earth’s magnetosphere, we give a brief summary of magnetic
reconnection, and shortly present the Cluster and the new Magnetospheric
Multiscale (MMS) missions. After that an extensive introduction to magnetic
nulls is given and a summary of our recent findings concerning them and the
reliability method we created. Thereafter, an introduction to particle acceleration and plasma heating, an important part of our study in Paper II, is given
where a short section regarding current sheets is included. In the last chapter
we give a short look into future work.
1
2. Basic plasma physics
In order to understand the detailed studies given in Paper I and II some essential concepts needs to be introduced. Basic plasma physics is a wide subject,
described in great detail in textbooks such as [12, 4, 31, 5, 39, 37]. This chapter gives a short introduction to the most basic concepts of plasma.
2.1 Plasma
In our daily lives, matter is usually present in three different states: gas, liquid,
and solid. How we can manipulate and describe these states has shaped how
we scientifically view the world. The further from Earth’s surface we look the
more different the environment is. At about a height of 80 km from the surface
the atmosphere starts to contain a large fraction of ionized particles, which is
the start of the ionosphere [37]. Going even further into space, high above the
ionosphere, almost all matter is ionized due to the electromagnetic radiation
from the Sun (Fig. 2.1). This introduces a new fourth state of matter, so called
plasma, which is a gas of charged particles that dominates large volumes of
the Universe. The nearest regions to us dominated by plasma are the Earth’s
ionosphere and magnetosphere.
2.2 Characterization
In most environments, including the near Earth space, plasma consists of negative electrons and positive ions. There can be several ion species present, such
as protons, oxygen ions, etc. Each of the plasma species can be characterized
by its temperature, T, and number density, n. Temperature 1 in space physics
is defined as an average kinetic energy of particles in the reference frame moving with the average particle velocity. Plasma in the outer magnetosphere is
collisionless and different plasma species can have different temperatures due
1 It
is standard in space physics to use electron volt (eV) as the unit to measure plasma temperature or any other energy quantity. The relation between energy, E , expressed in Joules and
temperature, T , expressed in K, Te [K], or in eV, Te [eV], is given by:
E = kb Te [K] = eTe [eV],
(2.1)
where kb is the Boltzmann constant and e is the elementary charge. From Eq. 2.1 we obtain
conversion factor for temperature, 1 eV = 11600K.
2
Figure 2.1. Artist rendition of the Sun and Earth relationship. Credit: NASA/Steele
Hill.
to different physical processes heating them. However, the charge density of
negative particles is always very close to the charge density of positive particles, which means that the plasma is quasi-neutral. Any deviation from the
quasi-neutrality will generate strong electric fields which works to restore the
quasi-neutrality.
While temperature and density of plasma species are fundamental parameters characterizing plasma, there are many other important parameters. For
example, magnetic field strength and the ratio between plasma pressure and
magnetic field pressure (plasma beta) are important parameters controlling the
physical processes in the plasma. Electric fields can be present in plasma,
which affect the motion of charged particles. Particle distribution functions
can be non-Maxwellian, there can be anisotropies with respect to the magnetic
field, there can be large scale gradients in the plasma, there can be different
plasma waves present, etc. All this makes plasmas a very complex environment for studies.
2.3 Electromagnetic Interactions
Since plasma is made up of charged particles, electromagnetic interactions are
important. Electromagnetic interactions are controlled by a set of combined
equations that was introduced by James Clerk Maxwell(1931-1879) in 1873.
3
Today these equations are commonly referred to as Maxwell’s equations:
r
e0
—·B = 0
∂B
—⇥ E =
∂t
—·E =
(2.2)
(2.3)
(2.4)
— ⇥ B = µ0 J + µ0 e0
∂E
∂t
(2.5)
where E and B are the electric and magnetic field, respectively. J = e(ni vi
ne ve ) is the total current density, µ0 is the permittivity of free space, r =
e(ni ne ) is the total charge density, and e0 is the vacuum permittivity. Equation 2.2 can be interpreted as the electric field produced by electric charge,
where the electric field diverges (converges) near positive (negative) charge.
Equation 2.3 states that B is divergence free, in other words there are no magnetic monopoles. Equation 2.4 means that a varying magnetic field will produce a circulating electric field or vice versa. Equation 2.5 essentially says
that if you have a current or a time varying electric field, a circulating magnetic field is produced or vice versa.
If we assume that the electric and magnetic fields are related to the current
in the plasma through Ohm’s law with the electric resistivity, h,
hJ = E + v ⇥ B,
(2.6)
and neglect the last term on the r.h.s. in Ampére’s law (Eq. 2.5), then Faraday’s
law (Eq. 2.4), can be re-written using these two Eq. as [44, 49, 37]:
∂B
∂t
= — ⇥ (v ⇥ B) +
1 2
— B.
µ0 s
(2.7)
This equation is commonly referred to as the induction law. It shows how the
magnetic field, B, evolves with time, where the first term on the r.h.s. is the
advective term and the second term is the diffusion term. The ratio between
the advective and the diffusion term is called the magnetic Reynolds number,
⇥ ⇥
Rm = |—⇥1 (v⇥2B)| . The advective term is from where the saying the field lines
µ0 s —
B
are "frozen-in" to the plasma comes from, because it describes how the field
lines, B, are carried along the plasma, v. In other words, plasma elements
on one field line remains so at all times. Most of the plasma in the Universe,
except for stellar interiors, has Rm >> 1. This means that the diffusion term in
Eq. 2.7 can be neglected and we have a so-called ideal plasma. If the plasma
is ideal then plasmas of different origins on different magnetic fields can form
boundaries and touch, but are not capable of mixing unless some additional
resistivity is added to Ohm’s law [56].
4
f(v)
v
0
Figure 2.2. Illustration of an Maxwellian distribution function, the most common
theoretical distribution function.
2.4 Kinetic theory
In a kinetic description of plasma, particle distribution functions, f = f (r, v,t),
are used (Fig. 2.2). A particle distribution function gives the probability density of finding a particle with velocity v at a point r at the time t. Different
characteristic parameters of plasma, such as n, v, P and T, can be determined
by calculating different moments of the distribution function:
n=
P=m
Z •
•
hvi =
f (v)(v
Z •
Z
1 •
n
(2.8)
v f (v)d 3 v,
(2.9)
hvi)d 3 v,
(2.10)
•
•
hvi)(v
f (v)d 3 v,
T=
P
,
nkb
(2.11)
where kb is the Boltzmann constant and both T and P are tensors. The scalar
temperatures and pressures can be determined from taking the trace of the respective tensors. However, many physical problems cannot be solved by only
looking at the moments of the distribution function. Instead the full distribution function and its evolution is needed. A kinetic description of a plasma
is given by describing this evolution. The simplest possible form of equation
describing the evolution is derived by assuming collisionless plasma and is
called the Vlasov equation:
∂f
e
∂f
+ v · — f + (E + v ⇥ B) ·
= 0,
∂t
m
∂v
(2.12)
where e is the charge. The Vlasov equation can be interpreted as f (r, v,t)
is constant along the particle’s orbit in space. Many important problems in
plasma physics, such as particle acceleration, require a kinetic description of
the plasma.
5
3. Magnetosphere
The term magnetosphere refers to a space surrounding a planet where the
planet’s magnetic field controls the motion of the space plasma particles. The
magnetosphere plasma consists of electrons and ions (mainly protons) originating from the ionosphere and solar wind [4]. The outer boundary of the
Earth’s magnetosphere is called the magnetopause. It separates the interplanetary magnetic field (IMF) originating from the Sun and the Earth’s magnetic
field (geomagnetic field). The geomagnetic field prevents almost all of the
solar wind plasma from entering the magnetosphere and maybe later even our
atmosphere. The direction of the IMF at Earth varies due to solar eruptions
and velocity variations of the solar wind [39]. Figure 3.1 shows an illustration of the main components of the magnetosphere: the magnetopause, the
cusps, the plasmasphere, the plasma sheet, the magnetosheath, and the magnetotail with its tail lobes. Figure 3.1 gives a false impression of stationarity
of the magnetosphere. In reality, the magnetosphere is highly dynamic due
to variations of the solar wind. Due to the interaction of the geomagnetic
field and the solar wind the magnetosphere forms its characteristic extended
magnetotail[4]. If we did not have the solar wind the magnetosphere would
Figure 3.1. 2D sketch of the Earth’s magnetosphere. Credit: ESA/C. T. Russell.
6
Figure 3.2. Sketch of a dipole magnetic field around the Earth. The yellow shaded
area represents the day side while the grey represents the night side.
look like a perfect dipole like in Fig. 3.2. Upstream of the magnetosphere
a bow shock is formed where the supersonic solar wind is slowed down to a
subsonic speed. The polar cusps form at high latitudes and for southward IMF
separates closed magnetic field lines from the open field lines pulled away by
the solar wind to the magnetotail. The cusps are important because they are
the weak spots of the magnetosphere, the places where plasma and particles
from the solar wind can directly penetrate into the magnetosphere. The lobes
are regions with low density and open field lines: one end is connected to the
solar wind while the other is connected to the Earth. Most of the magnetotail’s
hot plasma is located in the 10 Re (Earth radii ⇠ 6378 km) thick plasma sheet
[4].
The conditions inside and between these regions determine how the solar
wind particles enter our magnetosphere. That is why one of the most important
parts to study in the magnetosphere are the boundary layers that separates
these different regions. Many different processes such as particle acceleration,
energy conversions, and plasma transport processes occur here.
7
4. Magnetic Reconnection
As mentioned before, in most part of the Universe the magnetic field is frozen
into the plasma. However, in some localized regions magnetic reconnection
(Fig. 4.1), a fundamental process, occurs which breaks this condition and allows plasma to mix [49, 48, 7]. In addition, reconnection allows magnetic
energy to be converted into particle acceleration and heating of the plasma.
Magnetic reconnection has been observed, or has been suggested to be present,
in the chromosphere [34], the solar wind [28], Earth’s magnetosphere [32, 43,
51, 60, 46], galaxies [59], comet tails [36], and even on other planets,for example, Saturn [1]. However, there are still many unanswered questions related
to the physics of magnetic reconnection.
A sharp change in the magnetic field, a so called shear, is needed for reconnection to occur, which by its very definition implies the existence of a
region of strong current (due to Ampére’s law). An electric field, to break the
frozen-in condition, is also required for reconnection to occur. As reconnection proceeds, plasma jets are formed, plasma is heated, strong currents are
generated, and many other processes are taking place. How exactly the reconnection electric field is generated is an open question. The resistive Ohm’s law
(Eq. 2.6), as assumed in the traditional 2D reconnection theories, is generally
not enough to break the frozen-in condition in collisionless space plasma [7].
This is because the resistive term is generally too small and is negligible compared to other contributing factors. Instead, it can be non-gyrotropic electron
distributions or anomalous resistivity due to plasma waves that allow the setup
of the reconnection electric field. Understanding all these processes is one of
the major goals of MMS, as well as other space missions such as Cluster.
8
Z
Bx> 0, Bz> 0,
V x> 0
X
Y
i e
Inflow Region
Outflow
Region
Outflow
Region
Bx< 0, Bz> 0,
V x> 0
Bx> 0, Bz< 0,
Vx< 0
i
e
Inflow Region
Bx< 0, Bz< 0,
Vx< 0
Figure 4.1. Illustration of the 2D reconnection diffusion region, where the frozen in
condition breaks down. The blue/white background gives the magnetic field strength
where the fading of the colors represents lower magnetic field strength. The magnetic
field lines are given by the black arrowed lines and the large grey arrows gives the ion
path through the diffusion region.
9
5. The Cluster mission
Cluster (Fig. 5.1) is a four spacecraft mission run by the European Space
Agency (ESA). The four spacecraft were launched a month apart into a polar orbit on 16 July, 2000 and 9 August, 2000 with an apogee and perigee of
about 19 and 4 Re, respectively [20]. Cluster is still an active mission and
has now been in space for 16 years. The operational design of having the
four spacecraft form vertices of a regular tetrahedron meant that for the first
time measurements in three dimensions, with the ability to distinguish between temporal and spatial changes, was possible. The evolution of the orbit as
well as the possibility of changing the separation between the spacecraft made
it possible for Cluster to investigate new regions of the Earth’s plasma environment. The main goal of the Cluster mission is to study three-dimensional
plasma structures. To do that each Cluster spacecraft carry an identical set of
11 instruments, which includes electric and magnetic field instruments as well
as particle instruments. Details on different kinds of discoveries made with
Cluster can for example be found in [21].
The sampling modes of the spacecraft can change, depending on the study
object. More telemetry for shorter periods, usually referred to as burst mode,
is one of those sampling modes. Periods of burst modes are either scheduled
based on the regions where something is expected to occur, for example, in
the magnetotail or at the magnetopause, but it can also be triggered by an
instrumental signal, such as a high-amplitude electric field. During burst mode
the magnetic field is sampled at 67.3Hz (15 ms) instead of the normal 22.4Hz
(45 ms) rate [3]. This means that if you have a very fast or thin structure only
the burst mode allows detailed studies.
In our study in Paper I we only used magnetic field measurements from the
fluxgate magnetometer (FGM) [3].
Figure 5.1. Artist rendition of the Cluster mission. Credit: ESA.
10
6. The Magnetospheric Multiscale mission
The Magnetospheric Multiscale (MMS) mission (Fig. 6.1) is a four spacecraft
mission run by the National Aeronautics and Space Administration (NASA).
The spacecraft were launched into a highly eccentric, equatorial orbit on March
12, 2015 [8] to investigate the very small region where the electrons decouple
from the plasma, commonly referred to as the electron diffusion region (EDR)
predicted by 2D reconnection theory. To do that the MMS spacecraft carry
an identical set of 16 instruments, including particle detectors, electric and
magnetic field instruments.
The three main goals of the mission are to: determine the role of turbulent
dissipation and electron inertial effects in EDR, determine the parameters that
control the reconnection rate and what the rate actually is, and lastly determine
the role the ion inertial effects have on reconnection.
Because the electron diffusion region is predicted to be very small and reconnection regions are generally fast moving the spacecraft instruments and
their orbit had to be designed in such a way that the spacecraft could hover
near the expected regions of reconnection with a high enough measuring rate.
For example, an electron diffusion region moving with 50 km/s with a typical
width of 5 km would be crossed by a spacecraft in only 0.1 s. Thus, the time
resolution of 0.03s at which the electron distribution functions are measured
allows us to obtain at least three distribution function measurements during
the crossing. Similarly, using the length scales of the ion diffusion region the
time resolution for the ions of 0.15s allows up to 30 measurements inside the
ion diffusion region. Thus, both electron and ion distribution functions can be
well resolved within their respective diffusion regions.
To get the high temporal resolution of particle distribution functions from
the Fast Particle Investigation instrument on MMS, eight sensors were placed
around the spacecraft. This allows us to measure all directions independent
of the spacecraft spin, unlike the Cluster mission where the 3D particle distributions are constructed using data from a full spin of the spacecraft [22, 16].
With such a high sampling rate of the distribution functions only about 20
min of burst data per day can be downloaded through the Deep Space Network (DSN) and the memory on-board each spacecraft can only handle 3 days
worth of data. The rest of the data is averaged down to a fast-survey rate
where the time resolution of the full distribution is 4.5s, which is comparable
to previous missions such as Cluster with a resolution of 4s [22, 16].
The magnetic field is sampled every 10 ms with an accuracy of 0.1 nT and
the electric field is sampled every 1 ms with accuracy better than 1 mV m 1
[55].
11
Figure 6.1. Artist rendition of the Magnetospheric Multiscale (MMS) mission. Credit:
NASA.
Due to the size of the diffusion region, the size of the burst mode data, and
the capabilities of the on-board memory the orbit of the spacecraft were chosen so that the apogee are near the expected reconnection sites, 12 Re on the
dayside and 25 Re on the nightside of the magnetosphere [27]. The spacecraft separation will vary between 10 to 400 km, since the optimum separation
to find the EDR is not known. The lowest range (10-160 km) was used at
the dayside, while a higher range (30-400 km) will be used on the nightside
magnetosphere. The burst data for all instruments are only measured in a region of interest determined by a model predicting the highest probability of
encountering reconnection regions [27]. However, due to the large sampling
and memory capabilities a scientist in the loop (SITL) need to go through the
quicklook survey data and determine which time intervals should be downlinked by setting their priorities.
In our study in Paper II we use particle data from the Fast Plasma Investigator (FPI) [47], magnetic field from the Fluxgate Magnetometer (FGM) [52],
and electric fields from FIELDS Electric Double Probe (EDP) [41, 18].
12
7. Magnetic Nulls
Magnetic nulls, regions of vanishing magnetic field, are important sites of energy release and possibly particle acceleration. Magnetic nulls, both as pairs
and single occurrences, have been observed in reconnection current sheets in
the Earth’s magnetotail [61, 60, 33, 17, 58]. They have been determined to
be the driving force of the Bastille flare (Fig. 7.1), one of the most eruptive
events seen on the Sun [2]. Other solar events like solar jets [62], brightening of a flare [14], and CME’s [42] are also believed to be connected with
reconnection at 3D nulls. Magnetic nulls have also indirectly been found in
abundance in the corona [23]. The magnetic nulls are possible sites of particle
acceleration. Near them plasma particles become unmagnetized, due to the
low magnitude of the magnetic field strength, and can be accelerated to high
energies by traveling along an electric field (see section 8).
7.1 Description of Magnetic Nulls
A magnetic null is a point in space where the magnetic field vanishes. A null’s
skeleton, a common term used when talking about nulls, refers to the topology
of the magnetic field lines near a 3D magnetic null [15, 40]. The skeleton is
separated into two structures (Fig. 7.2): the fan plane where several field lines
either flow out or into the null, and the spine where two field lines either flow
in or out of the null. The direction of both structures depends on which type
the null is. The fan acts as a "surface separatrix" separating two topologically
unique regions. The spine on the other hand is only a line and is therefore too
small to separate regions. In 2D these structures correspond to the red lines
seen in Fig. 7.3b). The skeleton can be found and re-created by assuming
linear magnetic field around the null using a first order Taylor expansion:
B(r) = —B · (r
rn ) ,
(7.1)
where r is the location in space and rn is the null position.
7.2 Different Types
In general, the eigenvalues, l1 , l2 , l3 , and corresponding eigenvectors of —B
(no matter which coordinate system it is in), defines the spine and fan of a
13
Figure 7.1. The Bastille Day solar flare. One of the most violent sun storms in history. The image was taken by the SOHO Extreme ultraviolet Imaging Telescope (EIT)
instrument in the 195 Å emission line. Credit: SOHO/NASA.
Spine
Fan Plane
Figure 7.2. Illustration of the skeleton of a 3D magnetic null.
3D null. Depending on the eigenvalues the nulls are either classified as A,
B, As, or Bs type [15, 29, 40] (Fig. 7.3). Two of the eigenvalues need to
either be complex conjugates of each other, with the third eigenvalue being
real, or all of the eigenvalues need to be real. This condition comes from the
Maxwell’s law stating that no magnetic monopoles can exist (Eq. 2.3). Thus,
the eigenvalues must satisfy the condition l1 + l2 + l3 = 0. The fan plane
is spanned by two eigenvectors corresponding to the eigenvalues whose real
parts have the same sign. Depending on if the eigenvalues spanning the fan are
complex or real, the field lines in the fan plane will either spiral about the null
point or radiate straight out, respectively. When the fan plane spirals the null
points are called spiral null points (As,Bs) while the other types are referred
to as radial nulls (A,B). The direction of the field along the spine is given by
the sign of det(—B) = l1 · l2 · l3 [40]. The different names A/As (referred to
as A-kind in Paper I) and B/Bs (B-kind in Paper I) come from the direction
of the field lines around the null point. A/As nulls have a positive det(—B)
value, which means that the magnetic field lines will diverge away from the
null point along the spine and converge toward the null point in the fan plane.
The other types of nulls, Bs and B, have the reversed direction of the magnetic
field with a negative value of det(—B).
A comprehensive mathematical study of 3D magnetic nulls was done by
Parnell et al. (1996) [45] where they categorized ideal and non-ideal nulls and
how they behave. This was done by rotating —B into the null’s coordinate
14
a)
b)
c)
d)
e)
f)
Figure 7.3. Illustration of the different types of nulls given by their topology. a) Otype null. b) X-type null. c)Bs type null. d) B type null. e) As type null. f) A type
null. a-b) are 2D null types while c)-f) are 3D nulls.
system:
—Bnull
0
1
= sµ0 @ 12 (q + jk )
0
1
2 (q
p
j?
jk )
1
0
A,
0
(p + 1)
(7.2)
where s is a scaling parameter unit [nT km 1 ] to make the parameters unitless. All the parameters given in tensor 7.2 define the topology of a null. For
example, a magnetic null is a spiral type (As/Bs) when jk > jth . p and q describe the potential (current free) part of magnetic field. j? , jk are the currents
perpendicular
and parallel to the spine of the magnetic null, respectively, and
p
jth = (p 1)2 + q2 is a threshold current derived by Parnell et al. (1996)
[45].
7.3 Location Methods
There are several ways to identify the location of magnetic nulls in spacecraft
data. One way is to cross it directly with a spacecraft. However, this is very
rare. Instead four spacecraft measurements are used to suggest the presence
of a null within a volume made up by the spacecraft. In Paper I we use the
two available multi-spacecraft methods to locate magnetic nulls using FGM
magnetic field data from Cluster. In this section we explain the Poincaré index
method in greater detail than given in Paper I and give a short summary of the
Taylor Expansion method.
15
Figure 7.4. Sketch of the concept of the Poincaré index method. The different color
lines represents the measurements taken by the different spacecraft.
7.3.1 Poincaré Index
Poincaré index (PI) is the most commonly used location method in space observations. It calculates the topology degree using a bisection method and
was introduced by Greene (1992) [30]. The method tests to see if there is a
magnetic null enclosed in a volume in configuration space (x,y,z) by mapping
the magnetic field values, at each time step, from the configuration space into
the magnetic field space (Bx , By , Bz ) (Fig. 7.4). If PI= ±1, the tetrahedron
encloses an odd number of magnetic nulls, while PI=0 means that an even
number of null points is enclosed. It is usually assumed that the spacecraft
tetrahedron is sufficiently small so that PI= ±1 means that only a single magnetic null is enclosed, and PI=0 means that no magnetic null is enclosed.
The PI is determined by first calculating the solid angle, Q, of each spherical triangle (Fig. 7.5) created from the vertices of the four spacecraft in the
magnetic field space, by assuming a linear magnetic field between the spacecraft. The solid angle can be calculated as the surface area on a unit spherical
triangle,
Q = A + B +C p,
(7.3)
where A, B, and C are the different angles in the spherical triangle (Fig. 7.5)
calculated using the cosine law:
✓
◆
1 cos(a) cos(b)cos(c)
A = cos
,
(7.4)
sin(b)sin(c)
✓
◆
1 cos(b) cos(a)cos(c)
B = cos
,
(7.5)
sin(a)sin(c)
✓
◆
cos(c) cos(b)cos(a)
C = cos 1
.
(7.6)
sin(b)sin(a)
16
Figure 7.5. Sketch of a spherical triangle. A, B, and C are the angles of the spherical
triangle and a, b, and c are the angles between the vectors of the origin.
a, b, and c are the smaller angles between the vectors around the origin given
by
a = cos 1 (VC · VB ),
b = cos 1 (VA · VC ),
c = cos 1 (VB · VA ),
(7.7)
(7.8)
(7.9)
where VA , VB , and VC are the unit magnetic field vectors given by three of
the vertices of the spacecraft tetrahedron making up one of the triangles of the
tetrahedron.
The PI is then calculated as the sum of the four solid angles, given by the
four triangular surfaces of the spacecraft tetrahedron, divided by 4pr2 , where
r=1 since it is a unit sphere. One important thing for this method to work is
that the sign for each solid angle is included when they are summed together.
If the spacecraft tetrahedron is not around zero in the magnetic field space then
the different areas, due to the sign, will take each other out and PI becomes
zero.
7.3.2 Taylor Expansion
The Taylor Expansion (TE) method, usually referred to as the First Order Taylor Expansion (FOTE) method [26, 24], is based on the Taylor equation (Eq.
7.1) used for recreating a null’s skeleton. Using positional and magnetic field
measurements from four spacecraft, the position of a potential null can be determined by taking the inverse of Eq. 7.1. The gradient of the magnetic field,
—B, is derived using four spacecraft measurements by assuming the magnetic
field changes linearly in space [9]. Thus, the gradient is assumed to be constant in space inside the spacecraft tetrahedron. It is only referred to as a potential null, since the equation will always give a position, and long distances,
17
Z
Y
X
Figure 7.6. Illustration of the volume used in Paper I to determine which magnetic
nulls are valid.
(r rn ), makes the linear field assumption invalid. In our study in Paper I we
only considered the position of a magnetic null as reliable if it was located
inside a box volume defined by the spacecraft location. The edges of the box
in each direction (x,y,z) were given by the maximum and minimum position
of the spacecraft (Fig 7.6) where the separation between the spacecraft had to
be smaller than the ion inertial length.
7.4 Statistical Study
In Paper I, we performed the first statistical study of magnetic nulls in the
Earth’s nightside magnetosphere. In this section we give a short summary of
the results presented in Paper I.
Figure 7.7 shows the location of all identified nulls in the Paper I study.
More nulls were located at the magnetopause than in the magnetotail current
sheet, due to the orbit and the dynamic nature of the magnetopause, resulting
in many crossings of the magnetopause. The TE method also found more nulls
than the PI method, which was expected since the box volume used with TE is
much larger than the tetrahedron used in PI. 80 % of the observed nulls were
type-identified as spiral nulls, which was very close to what we obtained when
forming a fully random magnetic field, suggesting that the physical processes
behind null formation do not favour any particular types.
7.5 Reliability Method
Spacecraft measurements usually suffer from problems such as instrument
noise, calibration issues etc. It is therefore important to have a method for estimating what effect small magnetic field fluctuations will have on the accuracy
18
140
15
120
80
0 Magnetopause
60
-5
40
-10
20
-15
Magnetotail Current Sheet a)
Nulls (TE)
Nulls (PI)
5
Z [Re] GSM
Dwell time [h]
100
5
0
140
120
100
80
0
60
Dwell time [h]
Y [Re] GSM
10
40
-5
20
b)
-5
-10
-15
X [Re] GSM
0
-20
Figure 7.7. The results from the statistical study done in Paper I. Each symbol gives
the position of the magnetic nulls found for both the Poincaré index (PI), green circle,
and Taylor Expansion (TE) method, red triangle, in Geocentric solar magnetospheric
(GSM) coordinates. The gray background gives the dwell time of the spacecraft in
each of the spatial bins. Credit: Eriksson et al. (2015) [19]
19
of the type identification of magnetic nulls, since it relies on the assumption
of magnetic field linearity. When using Cluster and MMS spacecraft data the
largest magnetic field disturbances originate from local plasma processes (e.g.,
localized structures on spatial scales smaller than the spacecraft separation or
waves), but can also be due to instrumental errors. In Paper I, we present our
method of estimating how reliable the type identification is. In this section we
give a brief summary of the method.
To create the method we used the Parnell et al. (1996) [45] method to transform —B into the null’s coordinate system to get the parameters that defines
the null’s topology (see section 7.2). The basic concept of the method is to
compare theoretical minimum disturbances capable of altering the type of the
null with typical magnetic fluctuations observed in the data. Examples of how
this method works can be found in Paper I.
There are two ways a magnetic null type can alter: it can either shift between A(As) or B(Bs), or from/to a spiral type. Using Ampéres law, the theoretical minimum disturbance required to alter a null type to/from a spiral type
is,
d B1 = µ0 sL( jk jth ),
(7.10)
where L is the characteristic separation between the spacecraft. Using the fact
that the sign of det(—B) determines whether the magnetic null is of A(As)
or B(Bs), the theoretical minimum disturbance required to alter a null type
between A(As) and B(Bs) is
d B2 = min (|Bi j · (Bik ⇥ Bil )|/|(Bik ⇥ Bil )|) ,
(7.11)
where d B2 can also be interpreted as the minimum of the inverse of a reciprocal magnetic field vector, i, j, k, l are arbitrary permutations of the four
spacecraft (1, 2, 3, 4), and Bi j = B j Bi .
20
8. Plasma Heating and Particle Acceleration
The heating of plasma to millions of degrees and the acceleration of charged
particles to high energies, well above thermal energies, have been observed in
many astrophysical plasma environments. How and where the plasma heating
and particle acceleration occur is in many cases still an open question. Particle
acceleration has been observed indirectly for solar flares [11], and directly in
the near Earth-space [50, 13, 25]. Several important mechanisms have been
shown to explain observed acceleration such as reconnection current sheets
[6], turbulence [50], and shocks [11]. However, the actual heating and acceleration mechanism involved, their importance and observations of them are
still in many cases unknown. The near Earth space is the most useful place to
study plasma heating and particle acceleration because it is easily accessible
for spacecraft; one can bring more instruments into the near Earth space and
therefore get more data back. The wealth of high-quality and high-resolution
data from the near Earth space, such as particle distribution functions and
electromagnetic fields, allows us to better determine the important heating and
particle acceleration mechanisms.
8.1 Basics Mechanisms
Plasma heating is defined as the increase of plasma temperature, while particle
acceleration is a more loosely defined process where some fraction of particles is accelerated to high energies. Particle acceleration can appear as well
resolved beams in distribution functions, but it can also show up as, for example, power law tails in the distribution functions. Some commonly observed
heating and acceleration processes are: betatron acceleration, Fermi-type acceleration, and wave-particle interactions. Observations of one acceleration
mechanism does not exclude the possibility of others also contributing to the
final energy gain. Thus, when investigating possible heating and acceleration
mechanisms a number of different combinations needs to be considered.
For particles to be accelerated the energy needs to come from somewhere.
In space when analyzing particle acceleration in most cases the gravitational
forces can be neglected and therefore the particles gain energy from electromagnetic forces. The Poynting theorem (the equation of energy conservation)
J·E =
∂ 1
1 2
( e0 E 2 +
B )
∂t 2
2µ0
—S,
(8.1)
21
indicates that energy is transferred from electromagnetic field energy into
plasma energy in regions where there is electric field and current. The first
term on the r.h.s of Eq. 8.1 gives the electromagnetic energy density, the secE⇥B
ond term S =
, the so called Poynting vector, gives the electromagnetic
µ0
energy flux, and the term on the l.h.s indicates if energy is received or given to
the plasma. For example, if J · E > 0 energy is given to the plasma. The work
done on a charged particle is given by
W = F · v.
(8.2)
Using the Lorentz force,
FL = e (E + v ⇥ B) .
(8.3)
We see that the acceleration of particles can only come from the electric field,
since the force due to the magnetic field is perpendicular to the velocity. Thus,
to gain energy a particle needs to move along E for all particle acceleration
mechanisms. However, the source behind the accelerating E for each mechanism is different. In Fermi and betatron the changes in the magnetic fields give
rise to the accelerating electric field due to Faraday’s law (Eq. 2.4). However,
for wave-particle interaction, if an electrostatic wave is experiencing Landau
damping then the electric field is given by the separation of charge density (Eq.
2.2). In the following subsections a brief summary of betatron acceleration,
Fermi acceleration, and Landau damping mechanisms is given.
8.1.1 Betatron Acceleration
Betatron acceleration is based on the conservation of the magnetic moment.
The first adiabatic invariant, the magnetic moment, is given by:
µ=
e?
,
|B|
(8.4)
where e? is the perpendicular energy. The magnetic moment is constant if the
particle motion is adiabatic, i.e. the temporal scale of the electric and magnetic
field changes observed by the particle is much larger than the gyroperiod (the
time it takes a particle to gyrate one orbit around a magnetic field line). What
this means is that if a particle drifts from a lower |B| to a higher one the
perpendicular energy must also increase (Fig. 8.1a). If there is no electric
field in the system, the energy of the particle should be constant, and therefore
the perpendicular energy increases as much as parallel energy decreases. This
for example explains particle mirroring in the dipole field. However, if there is
a time varying |B| in the system, there will be an electric field associated with
the magnetic field variations and during the period of magnetic field increase
both the perpendicular as well as the total velocity of the particle will increase.
22
a)
b)
c)
f(v)
E
B
x
ev
B1
x
B2
Vph
E
Vfield
e
0
v
Figure 8.1. Sketch of the different acceleration mechanisms mentioned in chapter 8.1.
a) Betatron acceleration where the blue lines are the magnetic field lines and the black
arrow gives the electron velocity. b) Fermi acceleration where the blue lines show the
magnetic field, the black arrow indicates the electron motion and Vfield is the velocity
of the convecting field lines. c) sketch of the Landau damping effect. If particles are
in the dark blue region they will gain energy from the wave while the particles in the
turquoise region will lose energy. The net effect on the wave will be a loss of energy.
Thus, the particle gains energy but the process is still reversible. However, the
state where e? >> ek is not stable and emissions of plasma waves, usually
whistler waves [38], will transport the energy from the perpendicular direction
to the parallel one. If this happens the energy gain will be irreversible and the
plasma is heated. Betatron acceleration is usually observed in space as an
increase in the plasma temperature perpendicular to the magnetic field (Fig.
8.2I).
8.1.2 Fermi Acceleration
Fermi acceleration is based on the conservation of the second adiabatic invariant, J. The second adiabatic invariant is given by:
J=m
Z
⌦ ↵
vk ds = 2ml vk ,
(8.5)
where l is the total length of the magnetic field line between two mirror points.
The second adiabatic invariant can only be conserved if the first adiabatic invariant holds true. The second invariant refers to the conservation of the back
and forth motion of a trapped particle between two mirror points on a magnetic
field line. What this means is that if a particle travels along a convecting field
line as in Fig. 8.1a the decrease in l will result in an increase in the average
parallel energy of the particle [6]. Thus, a particle can gain energy by bouncing back and forth on a convecting field line. Fermi acceleration is usually
observed in space as an increase of the temperature in the parallel direction to
the magnetic field (Fig. 8.2II).
23
I)
II)
a
f
b
g
c
h
d
i
e
j
Figure 8.2. Example of Fermi and Betatron acceleration seen in Space. I) Example of
Betatron acceleration observed on 1 October, 2007 by Cluster 1. The signature can be
seen in panel e). II) Example of Fermi acceleration observed on 3 September, 2006.
The signature can be seen in panel j). The different panels in the fig shows: a) and f)
Z component of the magnetic field in GSM coordinates. b) and g) the black curves
represents X component of the magnetic field in GSM coordinates and the blue lines
the plasma beta. c) and h) omnidirectional differential flux of the 4076 keV electrons.
d) and i) omnidirectional differential flux of the 0.3-20 keV electrons. e) pitch angle
distribution of the 4068 keV electrons and j) pitch angle distribution of 40-400 kEV
electrons. The grey area indicate the region where the accelerations were observed.
Adapted from Fu et al. (2011) [25].
24
8.1.3 Wave-particle Interaction
One example of wave-particle interaction that can increase energy is Landau
damping [54]. It occurs when some of the particles are in resonance with an
electrostatic wave; their velocity, v, is about the same as the phase velocity,
Vph , of the wave. If more of the particles have velocities slightly below the
wave phase velocity, the wave will damp and energy will be transferred from
the wave to the particles (Fig. 8.1c) by acceleration due to the wave’s electric
field. However, the opposite is also true. If more of the particles have velocities slightly above the phase velocity of the wave, then the particles will lose
energy and the wave will grow.
8.2 Current Sheets
Regions of strong current, usually with planar geometry referred to as current sheets, are important. Inside them magnetic energy dissipates to thermal
plasma energy and kinetic energy of accelerated particles [49, 7]. In situ observations have demonstrated that current sheets are formed in all kinds of
magnetospheres [35]. The most common observations of current sheets are in
the magnetotail plasma sheet. There the core part of the plasma sheet has a reversal of the magnetic field making it easy to observe current sheets (Fig. 3.1).
Previous observations [53] indicate that the dynamics of the tail current sheet
are driven by kinetic-scale effects and therefore depends on how thick the current sheet is. Many current sheets with kinetic scales have been found using
Cluster, where thin current sheets were found to be responsible for generating
50% of the magnetic fields total amplitude [53]. Thus, the dynamics of a very
small population of charged particles can be responsible for the configuration
of the whole magnetic field.
The duration of crossing a thin current sheet is about 0.5 s for typical velocities of current sheets in the magnetosheath [10], which is comparable to
the crossing time found in our study in Paper II. Thus, higher resolution than
the spin resolution given by Cluster is needed to resolve electron heating and
acceleration within a current sheet. The new MMS mission has an electron
resolution of 30 ms, which makes it possible to study narrow regions of strong
current in detail. Paper II presents a kinetic study of a thin and strong current
region and its related electron acceleration.
25
9. Looking Forward
Future studies include investigating the importance of the different particle acceleration mechanisms in 3D reconnection theories and what possible sources
could be behind them. Are magnetic nulls the most important sources of acceleration in reconnection? Are they regions of strong current? Or is something
else entirely more important? The new MMS mission with its higher time resolution data gives us the opportunity to seek the answers to these questions.
Furthermore, to better understand what we observe more collaborations with
the simulations community will be done.
26
10. Acknowledgments
I would like to thank my supervisors Andris Vaivads, Yuri Khotyaintsev, and
Stefano Markidis for the patience and support they have shown me. I would
also like to thank the FGM teams, the FPI team, the EDP team, the Cluster
Science Archive, and the MMS Science data center for providing the data for
these studies.
27
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