3D Magnetic Nulls and Regions of Strong Current in the Earth's Magnetosphere

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

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 pro- cesses, 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 in- side boundaries separating plasmas with di↵erent magnetic field orientations.

For the first time, a statistical study of magnetic nulls in the Earth’s night- side 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 Wal´en

relation suggesting that plasmas on both sides of the current region are mag-

netically connected. We speculate on the formation mechanisms of the current

sheet and the physical processes inside and around the current sheet.

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

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), Iden- tifying 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),

Contents

1 Introduction . . . . 1

2 Basic plasma physics . . . . 2

2.1 Plasma . . . . 2

2.2 Characterization . . . . 2

2.3 Electromagnetic Interactions . . . . 3

2.4 Kinetic theory . . . . 5

3 Magnetosphere . . . . 6

4 Magnetic Reconnection . . . . 8

5 The Cluster mission . . . . 10

6 The Magnetospheric Multiscale mission . . . . 11

7 Magnetic Nulls . . . . 13

7.1 Description of Magnetic Nulls . . . . 13

7.2 Different Types . . . . 13

7.3 Location Methods . . . . 15

7.3.1 Poincaré Index . . . . 16

7.3.2 Taylor Expansion . . . . 17

7.4 Statistical Study . . . . 18

7.5 Reliability Method . . . . 18

8 Plasma Heating and Particle Acceleration . . . . 21

8.1 Basics Mechanisms . . . . 21

8.1.1 Betatron Acceleration . . . . 22

8.1.2 Fermi Acceleration . . . . 23

8.1.3 Wave-particle Interaction . . . . 25

8.2 Current Sheets . . . . 25

9 Looking Forward . . . . 26

10 Acknowledgments . . . . 27

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 de- tailed 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, unit- less parameters characterizing those plasma environments, such as the plasma beta, can be very similar to each other [57]. Thus, studying processes occur- ring 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 happen- ing in space. Simulations results are commonly used by comparing them to space observations. Powerful computer simulations can test out different the- oretical 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 mag- netosheath 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 acceler-

ation 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.

2. Basic plasma physics

In order to understand the detailed studies given in Paper I and II some essen- tial 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 chap- ter 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 nega- tive 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 mov- ing 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 temper- ature or any other energy quantity. The relation between energy, E , expressed in Joules and temperature, T , expressed in K, T _e [K], or in eV, T _e [eV], is given by:

E = k b T e [K] = eT e [eV], (2.1)

where k _b 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 par- ticles, 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 param- eters 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 environ- ment 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.

Today these equations are commonly referred to as Maxwell’s equations:

— ·E = r

e ₀ (2.2)

— ·B = 0 (2.3)

— ⇥ ⇥ ⇥ E = ∂B

∂t (2.4)

— ⇥ ⇥ ⇥ B = µ 0 J + µ 0 e 0 ∂E

∂t (2.5)

where E and B are the electric and magnetic field, respectively. J = e(n i v _i n e v e ) is the total current density, µ ₀ is the permittivity of free space, r = e(n _i n _e ) is the total charge density, and e ₀ is the vacuum permittivity. Equa- tion 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 mag- netic monopoles. Equation 2.4 means that a varying magnetic field will pro- duce 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 mag- netic 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

µ ₀ s — ² B. (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, R m = ^{|—⇥ ⇥ ⇥(v⇥} ₁ ^{⇥ ⇥B)|}

µ0s — ² B . The advective term is from where the saying the field lines 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 R _m >> 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

0

f(v)

v

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 den- sity 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 = ^{Z •}

• f (v)d ³ v, (2.8)

hvi = 1 n

Z _•

• v f (v)d ³ v, (2.9)

P = m ^{Z •}

• f (v)(v hvi)(v hvi)d ³ v, (2.10) T = P

nk _b , (2.11)

where k _b 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 re- spective tensors. However, many physical problems cannot be solved by only looking at the moments of the distribution function. Instead the full distribu- tion 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

∂t + v · — f + e

m ( E + v ⇥ ⇥ ⇥ B) · ∂ f

∂v = 0, (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.

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) orig- inating from the ionosphere and solar wind [4]. The outer boundary of the Earth’s magnetosphere is called the magnetopause. It separates the interplan- etary 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 illustra- tion of the main components of the magnetosphere: the magnetopause, the cusps, the plasmasphere, the plasma sheet, the magnetosheath, and the mag- netotail 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.

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 al- lows 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 ex- ample, 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 re- connection 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 reconnec- tion proceeds, plasma jets are formed, plasma is heated, strong currents are generated, and many other processes are taking place. How exactly the recon- nection 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 com- pared 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

Outflow Region Outflow

Region

Inflow Region

Inflow Region i e

i e

B

> 0, B

< 0, V

< 0

B

< 0, B

z

< 0, V

< 0 B

< 0, B

> 0,

V

> 0 B

> 0, B

> 0, V

> 0

X Z

Y

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.

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 po- lar 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 be- tween 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 envi- ronment. 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 re- connection 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 distri- butions 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 Net- work (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].

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 cho- sen 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 space- craft 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 re- gion 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 down- linked by setting their priorities.

In our study in Paper II we use particle data from the Fast Plasma Investi- gator (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 en- ergy 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], brighten- ing 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 r n ) , (7.1)

where r is the location in space and r n is the null position.

7.2 Different Types

In general, the eigenvalues, l ₁ , l ₂ ,l ₃ , and corresponding eigenvectors of —B

(no matter which coordinate system it is in), defines the spine and fan of a

Figure 7.1. The Bastille Day solar flare. One of the most violent sun storms in his- tory. 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 l ₁ + l ₂ + l ₃ = 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) = l 1 · l 2 · l 3 [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) O- type 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:

—B null = sµ ₀ 0

@ 1 ¹ ₂ (q j _k ) 0

1 2 (q + j _k ) p 0

0 j _? ( p + 1)

1

A , (7.2)

where s is a scaling parameter unit [nT km ¹ ] to make the parameters unit- less. 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 j _k > j _th . p and q de- scribe the potential (current free) part of magnetic field. j _? , j _k are the currents perpendicular and parallel to the spine of the magnetic null, respectively, and

j th = p

( p 1) ² + q ² 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.

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 ob- servations. 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 (B x , B y , B z ) (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 mag- netic 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 spher- ical 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 space- craft. 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:

A = cos ¹ ✓cos(a) cos(b)cos(c) sin(b)sin(c)

◆

, (7.4)

B = cos ¹ ✓cos(b) cos(a)cos(c) sin(a)sin(c)

◆

, (7.5)

C = cos ¹ ✓cos(c) cos(b)cos(a) sin(b)sin(a)

◆

. (7.6)

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 ( V _C · V B ), (7.7)

b = cos ¹ ( V _A · V C ), (7.8)

c = cos ¹ ( V B · V A ), (7.9)

where V _A ,V _B , and V _C 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 4pr ² , 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 Tay- lor 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 de- termined 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 con-

stant in space inside the spacecraft tetrahedron. It is only referred to as a po-

tential null, since the equation will always give a position, and long distances,

X

Y Z

Figure 7.6. Illustration of the volume used in Paper I to determine which magnetic nulls are valid.

( r r n ), 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 esti-

mating what effect small magnetic field fluctuations will have on the accuracy

18

-15 -10 -5 0 5 10 15

a)

Y [Re] GSM Dwell time [h]

0 20 40 60 80 100 120 140

X [Re] GSM

-20 -15

-10 -5

-5 0 5

b)

Z [Re] GSM Dwell time [h]

0 20 40 60 80 100 120 Nulls (TE) 140

Nulls (PI) Magnetopause

Magnetotail Current Sheet

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]

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 trans- form —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 be- tween A(As) or B(Bs), or from/to a spiral type. Using Ampéres law, the theo- retical minimum disturbance required to alter a null type to/from a spiral type

is, dB 1 = µ 0 sL( j _k j _th ), (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

dB ₂ = min(|B i j · (B ik ⇥ B il ) |/|(B ik ⇥ B il ) |) , (7.11) where dB ₂ can also be interpreted as the minimum of the inverse of a re- ciprocal magnetic field vector, i, j,k,l are arbitrary permutations of the four spacecraft (1,2,3,4), and B i j = B j B i .

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 accel- eration 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 parti- cles is accelerated to high energies. Particle acceleration can appear as well resolved beams in distribution functions, but it can also show up as, for exam- ple, power law tails in the distribution functions. Some commonly observed heating and acceleration processes are: betatron acceleration, Fermi-type ac- celeration, 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 electro- magnetic forces. The Poynting theorem (the equation of energy conservation)

J · E = ∂

∂t ( 1

2 e ₀ E ² + 1

2µ ₀ B ² ) —S, (8.1)

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 sec- ond term S = E ⇥ B

µ ₀ , the so called Poynting vector, gives the electromagnetic 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,

F L = 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 mecha- nism 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

B

e v B

E B

e V

0

f(v) V

a) b) c)

v

x

E

x

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 V field 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 _? >> e _k 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 invari- ant, J. The second adiabatic invariant is given by:

J = m ^Z v _k ds = 2ml ⌦ v _k ↵

, (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 in-

variant 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 bounc-

ing 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).

a

b

c

d

e

f

g

h

i

j

I) II)

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, V _ph , 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 veloci- ties 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 cur- rent sheets, are important. Inside them magnetic energy dissipates to thermal plasma energy and kinetic energy of accelerated particles [49, 7]. In situ ob- servations 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 re- versal 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 cur- rent 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 ve-

locities 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.

9. Looking Forward

Future studies include investigating the importance of the different particle ac- celeration mechanisms in 3D reconnection theories and what possible sources could be behind them. Are magnetic nulls the most important sources of accel- eration in reconnection? Are they regions of strong current? Or is something else entirely more important? The new MMS mission with its higher time res- olution 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.

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