Acceleration of High-Energy Electrons by Magnetic Reconnection in Coronal Nanoflares

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Acceleration of High-Energy Electrons by Magnetic Reconnection in Coronal Nanoflares

Helle Merethe Bakke

Master’s Thesis, Spring 2018

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Abstract

The heating of the solar corona is an open issue in solar physics. It is established that processes related to the magnetic field contributes to the energetics of the corona, but the details of the physical mechanism is still debated. In this thesis we investigate a heating mechanism by acceleration of high-energy electrons by magnetic reconnection in coronal nanoflares. Because nanoflares are difficult to observe directly, we approach the problem using numerical simulations. We present an analytical method of finding magnetic reconnection sites by studying magnetohydrodynamic descriptions and magnetic field topology. The developed method is implemented into a module created for the numerical MHD code Bifrost. The module locates electron acceleration sites based on the calculation of reconnection sites. The electron energies are assumed to be non- thermal, and can not be described by a Maxwell-Boltzmann energy distribution. We present a power-law distribution as a proxy for the spectrum of non-thermal energies, where the inital energy is calculated by considering the total distribution with both thermal and non-thermal components. We compare two Bifrost simulations, with and without the effect of accelerated electrons. Vertical velocity variations at locations of elevated electron beam heating are investigated, where we find evidence of chromospheric evaporation. This finding is further supported by line asymmetries in the Mg II h&k synthetic spectra, investigated due to their formation in the chromopshere.

In addition, we find brightenings in the line peaks, indicating heating of the chromospheric plasma. The developed method of finding reconnection sites is sufficient to describe locations of electron acceleration, and the beam heating simulation suggests evidence of heating signatures in the chromosphere.

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I would like to thank my supervisor Boris Vilhelm Gudiksen for giving me this interesting project, allowing me to explore aspects of the solar atmosphere both analytically and numerically. I would also like to thank Lars Frogner for the collaboration on this project. I have appreciated our academic discussions and occasional coffee breaks.

Thank you also to fellow students at the ITA, and especially to Andri for taking the time to correct my thesis from a cosmological point of view. Thank you Helene for your encouragement and positivity. Dearest of appreciation to my family for their con- tinuous support, with special thanks to Anette for creating the beautiful art on the front page of this thesis. Thank you Kristoffer Robin for your academic insights, and for proofreading this thesis. Finally, I am grateful to my partner Aleksander for being the biggest support I could ever have, and for always encouraging me to do my best.

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Introduction

The Sun is the primary source to life as we know it. This giant ball of plasma is continually fusing light elements into heavier ones, producing energy that is transported from the core to the surface. Outside the visible surface lies the solar atmosphere, divided into thephotosphere,chromosphere and corona. The solar atmosphere is dynamic and time-varying, and each layer has different properties such as temperature and density. The increase in temperature from a few thousand Kelvin in the photosphere to millions of Kelvin in the corona has raised questions for decades, and is known as the coronal heating problem. This is a major problem in solar physics that is continuously studied through observations and theoretical models, with a number of proposed ex- planations. We present a heating mechanism by magnetic reconnection in the corona, where electrons are accelerated to high non-thermal energies during solar flares.

1.1 Background and Motivation

The solar corona is heated by processes connected to the magnetic field, but it is not yet determined how these processes work. Energy exchanged between the corona and the lower atmosphere is observable in the chromosphere and transition region, as they are sensitive to heating mechanisms. Instabilities in the magnetic field produce reconnection of field lines, where energy is released in bursts called solar flares. They range from small brightenings to large, violent events with massive energy releases. A proposed mechanism for coronal heating is by magnetic reconnection in solar flares.

Small flares are more common than large flares, and it was proposed by Parker (1988) that nanoflares heated the corona. This was further investigated by Testa et al. (2014), who proposed that electrons accelerated by magnetic reconnection leave an observable signature in the chromosphere and transition region. They compared IRIS¹ observations of short-lived brightenings at coronal loop footpoints with numerical simulations of heating by non-thermal electron beams. Many of the brightenings observed in the Si IV spectral line were associated with blueshifts compatible with non-thermal electrons that deposit their energy directly into the chromosphere and lower transition region.

1Interface Region Imaging Spectrograph

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Numerical simulations with electron beams showed brightenings observed in several lines, among them Si IV and Mg II. The combination of observations and simulations can provide insight into the mechanism of non-thermal electron acceleration by coronal nanoflares.

Nanoflares are difficult to observe directly due to the limited accessibility of hard X-ray studies in such events. Hard X-rays contribute to the understanding of non-thermal electron properties, and the constraint results in limited knowledge of how non-thermal particles from nanoflares contribute to the heating of the corona. Potential heating signatures are obtained in regions that are responsive to heating. The corona is highly conductive, and shows no traces of heating signatures. The transition region is responsive to heating as the density and temperature gradients change rapidly, and the chromosphere shows signatures of where the non-thermal electrons deposit their energy.

In the latter, electrons lose their energy through collisions with the dense chromosphere described by a thick-target model (Brown et al., 2009).

Baumann et al. (2013) investigated particle acceleration in coronal 3D reconnection null-point regions, and found that a non-thermal population of electrons form in a simulated pre-flare phase. The simulations show a Maxwell-Boltzmann distribution combined with a power-law tail that is most likely a contribution from non-thermal electrons. The electrons are initially Maxwell-Boltzmann distributed, and randomly drawn electrons from this distribution are accelerated to higher, non-thermal energies.

They found that electrons are accelerated at sites where there is an imbalance between the current density and the curl of the magnetic field, as a consequence of reconnection processes.

1.2 Problem Statement

The aim of this thesis is to provide a better understanding of how magnetic reconnection accelerates electrons to non-thermal energies. The proposed mechanism is by magnetic reconnection in coronal nanoflares. Signatures from accelerated electrons should be visible in spectral lines that form in the chromosphere, where the electrons deposit most of their energy. Due to difficulties in observing coronal nanoflares, we investigate heating properties of accelerated electrons with a different approach. The numerical MHD code Bifrost is able to provide realistic models of the solar atmosphere in 3D. Simulations have provided results that are in agreement with observations, except in a few crucial ways. The emission lines originating in the solar chromosphere and transition region are too narrow and simple compared with the observed lines. This thesis describes the theory behind the creation of a new numerical module for the Bifrost code, written specifically for this project, and the first results it has produced. The module is now able to split the energy released in flares into two different types. One is the usual local heating, the other is non-thermal electrons. These electrons are then followed along

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1.3 Outline 3

the magnetic field until they impact in the transition region and chromosphere, where they deposit their energy. Two simulations, with and without accelerated electrons, are compared in order to determine the effect of electron beam heating. In particular, we investigate the Mg II h&k synthetic spectra by looking for signatures of electron impact in the chromosphere.

1.3 Outline

The structure of this thesis is as follows. We start by introducing the relevant background material in Chapter 2. This includes a brief overview of the Sun, its atmosphere, solar flares and magnetohydrodynamic descriptions. Chapter 3 gives a detailed description of the method developed to find magnetic reconnection sites, both analytically and numerically with Bifrost data. The thermal and non-thermal electron energy distributions are also presented in this chapter. In Chapter 4 we present our findings. This includes analysing the simulation atmospheres, interpreting the reconnection sites and inspecting Mg II h&k synthetic spectral lines. Chapter 5 contains a discussion of the results, and Chapter 6 gives a conclusion and suggestions for future work.

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Theory

2.1 The Structure of the Sun

A major problem in solar physics is the heating of the chromosphere and corona. The temperature varies dramatically between the different atmospheric layers, and there exists no definite explanation of this phenomenon. This section provides an overview of the solar interior and atmosphere, and introduces a heating mechanism central in this thesis.

2.1.1 The Solar Interior

The Sun is a nearly perfect sphere consisting of hot plasma that is being held together by its own gravitational attraction. It has massM= 1.99·10³³g, radiusR= 6.96·10¹⁰ cm, mean density ρ¯= 1.408 g/cm³, luminosity L = 3.84·10³³ erg s⁻¹ and effective temperature T = 5778 K. The solar interior comprises of hydrogen (92%), helium (8%) and a small fraction of heavier elements (0.1%). Helium (He) nuclei is mainly built from hydrogen (H) nuclei by the proton-proton chain reaction

4¹H→⁴He+ 2e⁺+ 2ν+ 26.7MeV (2.1) but a fraction is also built from the CNO cycle. The following is an example of a reaction producing helium nuclei by the last reaction in the CNO-I cycle:

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7 N+¹₁H→¹²₆ C+⁴₂He+ 4.97MeV (2.2) The Sun is divided into three regions, the core, radiative zone and convection zone.

Energy is generated by nuclear fusion in the core, and is transported by radiative diffusion across the radiative zone. The mean free path of photons is short due to the solar interior being opaque, and it takes many years for the radiation to cross the radiative zone. In practice it takes the photons 170,000 years to travel from the core to the surface of the Sun. In the convection zone, plasma cells carry heat as they rise, then cool down when reaching the surface. This leaves an observable granular pattern at

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2.1 The Structure of the Sun 5

0 5 10 15 20

x [Mm]

0 5 10 15 20

y [Mm]

Granular pattern in Bifrost simulation

0 5 10 15 20

Figure 2.1: Figure showing granular pattern in a simulation done with Bifrost. The height corresponds toz= 0.118Mm, which is in the photospheric layer.

the surface of the Sun, as illustrated in Figure 2.1.

2.1.2 The Solar Magnetic Field

The Sun’s magnetic field is responsible for all solar magnetic phenomena, such as sunspots, solar flares and solar wind. The origin of the solar magnetic field is unknown, but a popular theory is known as the solar dynamo. The theory is coupled to the solar cycle, each cycle lasting from one sunspot minimum to the following. Sunspots are believed to hold a strong azimuthal magnetic field generated in the solar interior. A toroidal field, a field whose lines of force are circles around the solar axis, is driven to the surface by convective buoyancy, giving rise to sunspots (Stix, 2004). Another signature of the global solar magnetic field is the solar corona. The coronal field, also referred to aspoloidal field, changes with the solar cycle, so that the coronal field is at a minimum when the sunspot field is at a maximum and reversing (Tobias, 2002).

The dynamics of the solar dynamo can be illustrated in Figure 2.2. A crucial ingredient for the solar dynamo action is the Sun’s differential rotation. In the late 1850’s, Richard Carrington observed sunspots rotating faster at lower latitudes, and concluded that the Sun rotated differentially. The differential rotation converts the poloidal magnetic field to toroidal field. Research on the solar interior has proved that a strong shear layer, called the tachocline, exists at the lower boundary of the convection zone, responsible for the generation of the toroidal field (Tobias, 2002; Priest, 2014). The strong shear is capable of storing the magnetic field, while the differential rotation strengthens the

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(a) (b) (c) (d) (e)

Figure 2.2: The solar dynamo. (a) The Sun’s differential rotation converts poloidal field to toroidal field at the tachocline region (inner circle). (b) Toroidal field around the tachocline. (c) Turbulence in the toroidal field. (d) Poloidal field is generated due to turbulence. (e) Large-scale poloidal field.

field. The reconversion from toroidal to poloidal field is debated, but a theory is that the toroidal and poloidal fields are generated in nearby regions such as the tachocline.

As the toroidal field rises, it may be shredded by convection, generating a poloidal field.

If the toroidal field is strong it may rise without being shredded, and a poloidal field is not generated. This is consistent with the coronal field being at a minimum when the sunspot field is at a maximum, and reversed. Hence, the cycle is complete. For a detailed analysis on the solar dynamo, see Tobias (2002).

2.1.3 The Solar Atmosphere

The solar atmosphere extends far beyond the observable surface, and is defined as the part of the Sun from which photons can escape into space. It consists of three regions with different properties such as temperature, thickness and optical depth. Optical depth is a convenient measure of transparency in the medium, and is defined as

I =I₀e^−τ (2.3)

whereI is the observed intensity along the beam of radiation,I₀ is the intensity of the radiation at the source andτ is the optical thickness. The fraction of the radiation that remains can then be represented asI/I0 =e^−τ, and the development of the exponen- tial decay can be seen in Figure 2.3. The value of τ determines whether the medium isoptically thick (τ >1) oroptically thin (τ <1), and this value varies in the different atmospheric layers.

There exists a variety of different atmospheric models, and one of the best known is the VAL model (Vernazza et al., 1981). The model represents the average quiet Sun temperature distribution. It is derived from the EUV continuum, the Lαline and other observations, and the depths where the different continua and lines originate are spe- cified in the model. The solar atmosphere is inhomogeneous, dynamic and time-varying, causing properties like temperature and density to continually change as heated and cooled plasma moves due to different physical processes. Still, the VAL model is im-

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0 1 2 3 4 5

τ

0.0 0.2 0.4 0.6 0.8 1.0

I/I0

Remaining photon energy as a function of optical thickness

Figure 2.3: The fraction of radiation that remains in the medium is dependent of the value of τ. As τ increase, the medium is getting more opaque. On the opposite side, asτ decreases, the medium is getting more transparent.

portant as a standard, providing significant understanding of the solar atmosphere.

Photosphere

The solar photosphere is a relatively dense atmospheric layer located above the convection zone. It is a few hundred kilometers thick, and emits most of the solar radiation.

Light of all wavelengths are emitted into space, and most of it passes through the overlying atmosphere without being absorbed. The particles that are absorbed in the chromosphere and corona creates dark absorption lines in the emitted spectrum, providing information on temperature, density, magnetic field strength and plasma motion.

The temperature is∼6600K at the bottom of the photosphere, decreasing to a minimum temperature of ∼ 4400K at a height of 500km. The optical thickness of the photosphere varies for different wavelengths, i.e. τ .1 in the visible, near-ultraviolet and near-infrared continua. The photosphere is optically thick in all except the weak- est spectral lines. Elements like Mg, Si and Fe provide most of the electrons in the photosphere (Grevesse and Sauval, 2002), increasing the fraction of ionised material.

The degree of ionisation varies with depth, but the fraction is overall small, making the photosphere only partially ionised.

The solar atmosphere is often incorrectly illustrated as spherical layers of different thickness. Due to convective motions, the photospheric layer is not a perfect sphere.

As previously mentioned, convection cells create granular patterns on the surface. The granules vary in size and brightness, making the photosphere non-uniform. Granular networks host different magnetic field strengths due to flux tubes from the solar interior.

The field strength of the flux tubes decrease with height, causing the magnetic pressure

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Figure 2.4: The VAL model. Average quiet-Sun temperature distribution taken from Vernazza et al. (1981). The figure indicates the approximate depths of where different spectral lines and continua originate.

to vary throughout the photospheric layer. Plasma beta, symbolised by β, is the ratio between the gas pressure and the magnetic pressure, and is defined as

β ≡ Pg

P_m = nkBT

B²/2µ₀ (2.4)

wheren is the density, k_B is the Boltzmann constant, T is the temperature, B is the magnetic field strength andµ₀is the vacuum permeability. Ifβ1, the magnetic field configuration is dominating the plasma motion. Ifβ 1, the magnetic field is moved with the bulk plasma flow. In the flux tubesβ <1, hence the magnetic field dominates the plasma motion. The photospheric regions around the tubes have β &1, and the gas pressure is dominating the magnetic pressure.

Chromosphere

The chromosphere lies above the photosphere, and can be observed during solar eclipses as a bright red color due to the emission of Hα. The layer is far thicker than the photosphere, and is on average a few thousand kilometers thick. As with the photosphere, it is optically thin in the visible, near-ultraviolet and near-infrared continua. Still, it is more transparent than its preceding layer. The temperature in the lower chromosphere

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starts at ∼ 4300 K¹, slowly reaching temperatures of ∼ 10⁴ K in the upper chromosphere. While the temperature slowly increases, the density decreases rather rapidly with height. Particles turn from neutral to ionised, changing the elemental abundances from the photosphere to the upper chromosphere and corona (Peter, 2002).

The chromosphere hosts interesting phenomena, such as filaments,prominences, spicules and a network outlined by supergranule cells. It is important to understand the structure, generation and behaviour of such phenomena to fully comprehend the com- plex dynamics of the chromosphere. Filaments are dark features suspended above the solar surface, and are formed in relatively cool and dense magnetic loops. Prominences are filaments observed at the solar limb, and have the shape of loops against the dark background. The filaments collapse if the supporting magnetic loops become unstable, causing eruptions that release the plasma. Supergranulation is formed in the photosphere by plasma that rises at the centre of each cell, moves outwards and falls at the boundaries (Priest, 2014). The chromospheric network is a bright pattern outlined by these boundaries. Spicules are plasma jets ejected from the supergranule boundaries along the magnetic field. They are separated into two categories that, among other properties, depend on origin and duration. Type I spicules are mostly seen in active regions, and have typical lifetimes of 3 to 10 minutes. Type II spicules also occur in active regions, but dominate in coronal holes² and the quiet Sun. Their duration is about 10 to180s, which is much shorter than type I spicules.

The chromosphere is continually studied through observations and simulations, but difficulties appear due to its complicated behaviour and inhomogeneity. Its variations with height causes the plasma to transition from partially to fully ionised, and the collisions are unable to completely couple the neutral and ionised components. This gives rise to a relative net motion between the particles, referred to as ambipolar diffusion (Khomenko and Collados, 2012). The heating of the chromosphere is not yet understood, but partial ionisation effects propose an interesting magnetohydrodynamic mechanism to this problem. Shelyag et al. (2016) suggest a heating mechanism due to wave dissipation caused by ambipolar diffusion, that can lead to a chromospheric temperature increase in magnetic structures. Another proposed mechanism ismagnetic reconnection (Section 3.1), which is studied in detail in this thesis.

Transition Region

It is common to refer to the transition region as the narrow layer between the chromosphere and corona. The plasma in the transition region is rapidly heating up or cooling down, producing a thin and irregular layer where the temperature varies from

∼ 10⁴ K to ∼ 10⁶ K (Judge, 2008). The transition region is not a static horizontal layer, but rather a representation of the plasma. It has been suggested that transition

1This is the minimum temperature of the VAL model, but the actual temperature is difficult to evaluate as the chromosphere is highly non-uniform and time-varying.

2Regions of open magnetic field in the corona that are cooler and less dense than its surroundings.

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region plasma may occur in spicular jets as chromospheric plasma heated to transition region temperatures. However, there are difficulties observing this, as spicules have to be observed against a dark background. The transition region is observed in EUV emission lines that are only accessible from space, adding to the difficulties of studying this particular region.

Corona

The corona is the outermost layer of the solar atmosphere, and contains mostly ionised gas at temperature of a few 10⁶ K. This optically thin layer can be observed during solar eclipses as a crown around the disk, hence the name corona (latin for ’crown’).

The corona is full of interesting phenomena, but one of the most spectacular events is called coronal loops. They are bright structures of hot plasma that flows along magnetic field lines. The loop ends are rooted in places of strong magnetic fields, such as sunspots and edges of active regions. They vary in size and duration, and multiple loops are normally present at the same time. A single loop can last about a day, while a system of loops may endure for several solar rotations. The corona is also believed to be responsible for solar flares, as magnetic fields, in which governs the evolution of the corona, are the only adequate source of energy supply.

2.2 Solar Flares

The heating of the chromosphere and corona is a heavily debated subject in solar physics. A theory is that magnetic reconnection in the corona releases energy bursts called solar flares. These energetic events accelerate particles to high non-thermal energies, possibly heating the lower layers of the atmopshere. This section gives an overview of the evolution and classification of solar flares, and introduces a proposed heating mechanism bynanoflares.

2.2.1 Evolution

Solar flares are sudden brightenings in the solar atmosphere. The energy released varies from ∼ 10²⁷ erg in the smallest observed events to ∼ 10³² erg in the largest events.

While the smallest events appear as simple brightenings, the largest events can eject plasma into space. During a solar flare, a large number of particles are accelerated to high energies. There exists a number of proposed mechanism for particle acceleration, and Figure 2.5 serves as an example of particle acceleration by magnetic reconnection.

As field lines in the corona reconnect, magnetic energy is converted into heat and kinetic energy, causing particles to accelerate. The energy released during the flare is carried to the lower atmosphere by high-energy particles or thermal conduction. The heating of the chromosphere causes it to fill the flare loops with hot plasma, and temperature and density increase. This is calledchromospheric evaporation. As the coronal plasma cools and drains back to the chromosphere, temperature and density decrease. This is

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2.2 Solar Flares 11

calledchromospheric condensation.

Figure 2.5: Simple illustration of magnetic reconnection. Field lines of opposite direction are pushed together into a diffusion region. As they reconnect, energy is released and particles are accelerated. The magnetic field configurations change.

2.2.2 Classification

A variety of radiation is emitted during a solar flare, such as soft X-rays (SXRs), hard X-rays (HXRs) and even gamma-rays (γ) for the most violent flares (Jeffrey, 2014).

The size of a solar flare is classified by its SXR flux. The letters A, B, C, M and X are used when classifying the flare, with an X-class flare being the largest. The classification depends on the peak flux measured byGOES³ at1AU, in the range 1-8 Å. X-class flares have a peak flux around 10⁻⁴ W/m², where the peak flux of each class is ten times greater than the preceeding one (Priest, 2014). SXRs are produced by bremsstrahlung from particles interacting within a high temperature plasma, while HXRs are produced by bremsstrahlung from electrostatic interactions of electrons with the background particles. Bremsstrahlung is produced by Coulomb collisions in the chromosphere and corona, and is the most important mechanism for the production of X-rays during solar flares. The chromosphere and corona are optically thin at high X-ray energies, and HXR studies from flares can provide information on the accelerated electrons, i.e. thermal or non-thermal energies.

2.2.3 Heating by Nanoflares

Parker (1988) proposed that the solar X-ray corona was heated by impulsive energy releases referred to as nanoflares. He analysed observations and suggested that the corona was a collection of nanoflares with an average energy of E . 10²⁴ erg. It is acknowledged that low energy flares are much more prevalent than large flares. The rate of occurence is not well established yet, mainly due to the difficulties in observing small flares. HXR observations of microflares with energyE ∼10²⁷ erg have revealed a presence of non-thermal particles in active regions (Testa et al., 2014). However, not all small-sized flares are accessible to HXR studies, in particular nanoflares which have

3The Geostationary Orbiting Environmental Satellites

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proven difficult to observe directly. By assuming that magnetic reconnection accelerates particles in both small and large flares, the effect of nanoflares should be visible in the chromosphere. The accelerated high-energy particles deposit a large fraction of their energy in the chromosphere, which creates a signature that should be observable in the Mg II h&k line profiles. These lines are formed at heights where accelerated electrons from nanoflares deposit their energy. The idea of high-energy particle acceleration by magnetic reconnection in nanoflares poses an interesting heating mechanism of the chromosphere and corona.

2.3 Particle Interactions in the Solar Atmosphere

The preceeding section introduced a mechanism of particle acceleration by magnetic reconnection in solar flares. As the accelerated particles travel to the lower atmosphere, they interact with the surrounding particles through Coulomb collisions. Particle interactions are important in order to determine the electron trajectories from acceleration to impact in the lower atmospheric layers. This section provides an overview of Cou- lomb collisions in general, and explains bremsstrahlung as the mechanism of X-ray emission.

2.3.1 Coulomb Collisions

The solar corona consists of fully ionised plasma, where electrons and ions interact by the electrostatic Coulomb force. This occurs via Coulomb collisions, as shown in Figure 2.6. Two charged particles interact through their own electric fields, and the particle trajectories change due to deflection by some angle θ. The simplest model describing Coulomb collisions takes an electron moving through a background of stationary ions.

Assuming the ions to be heavy with large atomic number Z, electron-ion collisions dominate over electron-electron collisions. As the electron approach the ion, it will slow down. In the electrostatic interaction potential energy is gained due to loss in kinetic energy, and at the time of reflection the entire initial kinetic energy is converted into potential energy. At this time the particles are at the distance of closest approach, b₀, and conservation of energy gives

1

2mv² = e²

4π₀b₀ (2.5)

for a collision between an electron and a stationary ion. e is the charge of the given particle and₀is the permittivity in vacuum. The distance of closest approach is readily calculated as

b₀ = e²

2π₀mv² (2.6)

for the same collision. The angle between the incoming and outgoing asymptotes is given by

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2.3 Particle Interactions in the Solar Atmosphere 13

m₁v₁

b m2

v2=0

v'₂ v'₁

ϴ1

ϴ2

m₁v1

b

m₂ v'₁

b ϴ

Figure 2.6: Coulomb collision between two charged particles. In the simple model describing Coulomb collisions, the particle of massm1 is the electron, while the particle of massm2is the stationary ion. Left: Both particles are deflected during the collision.

Right: The electron is deflected by a heavy, stationary ion.

cotθ 2 = b

b₀ =b2π₀mv²

e² (2.7)

where b is the impact parameter. The impact parameter is defined as the closest distance of approach by the particle, had it not been deflected during the encounter (Jeffrey, 2014). The largest possible value for the impact parameter is the Debye length, λ_D. This is a measure of a charged particle’s net electrostatic effect, and is characterising a shielding distance. The scattering cross section is found by considering the Rutherford formula, which is defined as the differential cross section per unit solid angle

dσ

dΩ = b²₀

4sin⁴^θ₂ (2.8)

A detailed calculation can be found in Appendix A.1. If the scattering angle is larger than90^◦, the cross section is calculated as

σ₉₀= 2π Z _π

π/2

dσ

dΩsinθdθ= 2π Z _b₀

0

bdb=πb²₀ (2.9)

where the relation between the differential cross section and impact parameter, ^dσ_dΩsinθ= bdb, is used. When the electron collides with the ion, it experiences a collisional drag force. The electron loses momentum in the direction parallel tov₁, which is transferred to the components of momentum perpendicular tov1. Eq. (2.10) is an effective cross section describing the average momentum transferred,

σ = 2π Z

(1−cosθ)dσ

dΩsinθdθ (2.10)

By using the Rutherford formula in Eq. (2.8), the above equation can be written as σ = 2π

Z θmax

θmin

(1−cosθ) b²₀

4sin⁴^θ₂sinθdθ (2.11)

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The Coulomb forceF =−e²/4π0r² is slowly decreasing because of the 1/r² variation of the potential. The decrease of the Coulomb force leads to a high probability of small- angle scattering (Lifshitz and Pitaevskii, 1981). By the small-angle approximation, the trigonometric functions can be written as

(1−cosθ)≈ θ² 2 4sin⁴θ

2 ≈ θ⁴ 4 sinθ≈θ

(2.12)

The cross section for small-angle scattering in a Coulomb field is then given by σ= 2πb²₀

Z θmax

θmin

θ² 2

1

θ⁴ 4

θdθ= 4πb²₀ Z θmax

θmin

1

θdθ (2.13)

Solving the integral gives the momentum transfer cross section σ = 4πb²₀ln

θ_max θmin

(2.14) The logarithm can be identified as the Coulomb logarithm, where

ln θ_max

θ_min

=ln b_max

b₀

=ln λ_D

b₀

=lnΛ_C (2.15)

It is seen from Eq. (2.15) that the maximum value of the impact parameter is equal to the Debye length. The momentum-transfer cross section is finally written as

σ= 4πb²₀lnΛC (2.16)

The Coulomb logarithm takes into account the total effect of all deflection angles ran- ging fromθ_mintoθ_max. Plasma physicists use that lnΛ_C is in the range of 1-10 (Pécseli, 2012), while in the flaring solar corona values around 20 is used (Jeffrey, 2014). In the numerical simulations done with Bifrost, Frogner (2018) has found this value to be around50.

2.3.2 Bremsstrahlung

Bremsstrahlung, or "braking radiation", is the most important X-ray emission mechanism during solar flares (Jeffrey, 2014). It is produced by Coulomb collisions in the chromosphere and corona, where electromagnetic radiation is emitted when an electron is slowed by surrounding particles. This can be seen in Figure 2.7. At the point of closest approach, the electron is slowed and deflected by the ion. The electron loses energy in the process, which is radiated as bremsstrahlung. If the energy of the electron is high enough, the radiation is in the X-ray part of the electromagnetic spectrum.

The mechanism of magnetic reconnection in solar flares accelerates electrons to high,

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2.4 Magnetohydrodynamics 15

non-thermal energies, producing HXR emission in the chromosphere. The energy of the electrons can be approximated by a power-law distribution. In general, this takes the form

f(E)∝E^−δ (2.17)

whereδis the spectral index. The non-thermal energy distribution is described in detail in Section 3.2.2.

v2

v₁

E₁ E₂

X-ray

+ -

-

Figure 2.7: An electron of energyE₁ and velocityv₁ is being slowed and deflected by a heavy, stationary ion due to Coulomb collision, producing X-ray bremsstrahlung. The trajectory of the electron changes, as well as the energy and velocity.

2.4 Magnetohydrodynamics

Magnetohydrodynamics, orMHD, is a framework for describing the dynamics of magnetic fields in electrically conducting fluids. Plasma is such a fluid, and its behaviour can be formulated by using Maxwell’s equations (2.18)-(2.21) together with the continuity equation (2.27), the equation of motion (2.28) and Ohm’s law (2.31). These equations will vary, depending on the plasma conditions. This section introduces equations central in this thesis, while providing an overview of two different MHD descriptions.

2.4.1 Basic Equations

Maxwell’s equations are a set of electromagnetic differential equations, and are written as

∇ ·E= ζ 0

(2.18)

∇ ·B= 0 (2.19)

∇ ×E=−∂

∂tB (2.20)

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∇ ×B=µ0J+µ00

∂

∂tE (2.21)

where ζ is the charge density, 0 is the vacuum permittivity and µ0 is the vacuum permeability. Gauss’ law (2.18) implies that electric charges give rise to electric fields, while Faraday’s law of induction (2.20) implies that time-varying magnetic fields give rise to electric fields. Gauss’ law for magnetism (2.19) simply states that the divergence of the magnetic field is zero. Ampere’s law (2.21) states that currents or time-varying electric fields can produce magnetic fields. The equation can be written in terms of normalising quantities to make an order of magnitude estimate for the different terms.

Introducing a characteristic length scaleα, a characteristic time scaleτ and characteristic amplitudes of fluctuations in electric and magnetic fields,⊥and β, Ampere’s law is written as

∇⁰×B⁰=µ0Jα β +

1 c²

_⊥ β

α τ

∂

∂t⁰E⁰ (2.22)

The prime denotes dimensionless quantities. An assumption of MHD is that the electromagnetic variations are non-relativistic, so that

˜

vc (2.23)

where˜v=α/τ is a characteristic plasma velocity. The assumption that ⊥

α ≈ β

τ (2.24)

provides the same order of magnitude on both sides of Eq. (2.20). This implies that the quantity ⊥/β is a characteristic velocity as well. The last term in Eq. (2.22), called Maxwell’s displacement current, has the magnitude

1 c²

_⊥ β

α τ

∂

∂t⁰E⁰ ≈ v˜²

c² ∂

∂t⁰E⁰ (2.25)

causing the term to be negligible in the MHD limit. Ampere’s law is obtained in its original form

∇ ×B=µ0J (2.26)

It can be argued that the omission of Maxwell’s displacement current is the basic assumption of MHD.

The continuity equation is written as

∂ρ

∂t +∇ ·(ρv) = 0 (2.27)

whereρ is the mass density and ρv is the flux of mass density. The mass density at a fixed point in space can vary for two reason. If an equal amount of fluid passes into and out of the volume, there is no gradient in the velocity and the fluid is incompressible

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(∇ ·v = 0). If there is more fluid passing into than out of the volume, there is no gradient in the mass density and the fluid is compressible (∇ ·v6= 0).

The equation of motion can be written as ρ

∂

∂t+v· ∇

v=−∇p+J×B+F (2.28)

where ∇p is the plasma pressure gradient, J×B is the force density and F denotes additional forces on the plasma, such as gravity and viscosity. The force density term comes from theLorentz force

F=q(E+v×B) (2.29)

whereq is the electric charge of a particle, v is its velocity, andE and B are external electric and magnetic fields, respectively. Under conditions of charge neutrality, the Lorentz force can be written in terms of the force density

f =ζ(E+v×B) =ζE+J×B (2.30) where ζ is the charge density. Plasma is a good conductor, and tiny deviations from the overall charge neutrality is expected to give rise to large currents. The forces due to these currents will dominate any ζE force densities, hence this term is neglected in the Eq. (2.28).

Ohm’s law is written as

J=σ(E+v×B) (2.31)

where σ is the electrical conductivity. Ohm’s law implies that the current density is proportional to the total electric field. An electric field is induced when plasma with velocityv is moving in the presence of a magnetic field. A second electric field contribution comes from E, which is present at all times in the absolute frame of reference.

A generalised form of Ohm’s law will be used further in this thesis, and is referred to as

E+v×B=R (2.32)

where R denotes the non-ideal effects of the plasma. Further investigation of these effects will not be discussed in this thesis. However, it is important to mention that a generalisation of Ohm’s law is appropriate when modelling the solar atmosphere. This is due to the use of a multi-fluid MHD model, which includes electrons, protons and neutrals in the MHD equations.

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An equation of state is present to relate variables that characterises the state of matter under a given set of physical conditions, such as pressure, temperature and volume.

The simplest equation of state is the ideal gas law

p=nk_BT (2.33)

wherenis the total number of particles per unit volume,kB is the Boltzmann constant and T is the temperature. Using n means to replace ρ by ρ = mn in the continuity equation and equation of motion. This is convenient when formulating a multi-fluid MHD description, as these equations are written for every component each. An equation of state is always present, as it is needed to close the set of equations.

2.4.2 Ideal MHD

In the limiting case of ideal MHD, it can be assumed that plasma is an ideal conductor.

Ohm’s law is altered by lettingσ→ ∞, and is expressed as

E=−v×B (2.34)

in order to keep the currents finite. The relation implies that there are no velocity components along the magnetic field caused by electric fields. The electric field in Faraday’s law can be eliminated by inserting the right-hand side of Eq. 2.34 into Eq.

(2.20), so that

∂

∂tB=∇ ×(v×B) (2.35)

For an incompressible flow (∇ ·v= 0), the continuity equation is written as

∂ρ

∂t +v· ∇ρ= 0 (2.36)

while the continuity equation for compressible flows follow Eq. (2.27). The current density in the equation of motion (2.28) can be eliminated by using

J= 1 µ0

(∇ ×B) (2.37)

from Ampere’s law (2.26). It is then written as ρ

∂

∂t+v· ∇

v=−∇p+ 1 µ0

(∇ ×B)×B+F (2.38)

In the ideal limit, magnetic field lines are frozen-in. This concept states that the flux through any material surface S following the fluid remains constant, meaning that it is possible to follow the motion of the magnetic field lines in space. The field lines are carried by the flow of particles. The individual plasma elements can move at different directions, but they remain magnetically connected via their mutually common field line. The magnetic flux is conserved, the magnetic field lines are conserved and the

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magnetic topology is conserved. The latter means that although the structure of the magnetic field is stretched and deformed, its topology remains unchanged. The concept of frozen-in field lines is an artifact. Magnetic field lines are usually drawn as aid, and it does not really make sense to follow their motion in time. However, in ideal MHD the particles act as identifiers, making it possible to do just that.

2.4.3 Resistive MHD

In the resistive, or non-ideal, MHD limit, the conductivityσis finite. Omh’s law follows Eq. (2.31), where the electric field can be expressed as

E= 1

σJ−v×B (2.39)

Using Eq. (2.37), the current density can be eliminated from the equation as E= 1

µ₀σ(∇ ×B)−v×B (2.40)

Again, the electric field in Faraday’s law can be eliminated by inserting (2.40) into (2.20), so that

∂

∂tB=∇ ×(v×B− 1

µ0σ(∇ ×B)) (2.41)

By using the differential expression∇ ×(∇ ×B) =∇(∇ ·B)− ∇²B together with Eq.

(2.19), Faraday’s law can be written as

∂

∂tB=∇ ×(v×B) + 1

µ₀σ∇²B (2.42)

where the last term is a diffusion term. This term causes the process to be irreversible.

Collisions between particles can cause their guiding centres to change, meaning that the particles change field line. Since the process is irreversible, it is no longer possible to follow the motion of the magnetic field lines in space. The frozen-in concept does not hold in resistive MHD, as the magnetic flux is not conserved.

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Method

3.1 Magnetic Reconnection

Magnetic reconnection is the reconfiguration of magnetic field lines in a plasma, and involves energy release and particle acceleration. It has been suggested that magnetic reconnection contributes to the heating of the corona through solar flares. As field lines reconnect, magnetic energy is converted into heat, kinetic energy and particle energy.

Magnetic reconnection in flares provide currents that accelerate electrons and heat the ambient plasma by joule heating.

The process of magnetic reconnection involves a change in the magnetic field line topology. Particles change from one field line to another, hence the frozen-in concept does not hold. When addressing magnetic reconnection, the limit of ideal MHD is not applicable. In the following we assume resistive, non-ideal MHD in order to find a criterion for reconnection in three spatial dimensions.

3.1.1 2D Reconnection

The simplest explanation of magnetic reconnection can be obtained by looking at the 2D case. Although this is not a realistic model, it provides insight into the theoretical aspects of reconnection mechanisms. Magnetic field lines of opposite directions are pressed together, and plasma flows out from between the two field directions. The field lines reconnect and spring out horizontally, creating an X-type shape of the configuration. This is called a separatrix surface, which separates regions of topologically different field lines (Biskamp, 2000).

2D reconnection can only occur at an X-point, and if E·B = 0. This requirement is due to the fact that the magnetic field vanishes at the X-type null point, and implies thatR⊥B, whereR is the right-hand side of the generalised Ohm’s law

E+v×B=R (3.1)

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3.1 Magnetic Reconnection 21

Since reconnection only occurs in non-ideal MHD,Rrepresents the non-ideal effects of the plasma. By settingR=δv×B, Ohm’s law can be written as

E+v×B=δv×B (3.2)

whereδv is singular at the null point (whereB= 0). A flux-preserving flow is always present, and is defined asw=v−δv. Eq. (3.2) can then be written as

E+w×B= 0 (3.3)

This equation implies that the flux transport velocity is given by w = E×B/B², which is smooth everywhere except at null points. It is singular at the null point, where magnetic field lines are being cut and rejoined. Because of this singularity, the field lines evolve as if they are reconnected at this point only (Pontin, 2012).

The Sweet-Parker Model

The first attempt to describe magnetic reconnection was done by Peter Sweet (Sweet, 1958) and Eugene Parker (Parker, 1957). They developed a model known as the Sweet- Parker model, where the concept of a resistive current sheet was introduced. They modelled a simple diffusion region of length 2L and width 2l between opposite field directions, see Figure 3.1. The model is restricted to steady state conditions,∂/∂t= 0, where the magnetic field is being pressed together and carried into the diffusion region at the same velocity as it is trying to diffuse outwards. The magnetic field lines merge with the velocity

v_i= η

l (3.4)

where η = 1/(µσ) is the magnetic diffusivity. The mass entering the diffusion region must equal the mass exiting at both ends, so that the density is uniform. The mass is conserved, and

4ρLv_i = 4ρlv_o

The fluid exiting the diffusion region has the outflow velocity vo= Lvi

l (3.5)

which gives an expression for the inflow velocity v_i =

ηv_o L

1/2

(3.6) The fluid at the null point is accelerated by the Lorentz force to a distance L where the fluid velocity is vo. The magnetic pressure PB =B_i²/2µ0 squeezes the fluid along

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2l

2L v_iB_i

v₀B₀ Diﬀusion region

Figure 3.1: Sweet-Parker reconnection model. Magnetic field lines are indicated with direction above and below the diffusion region.

the lines of force. Assuming incompressible flow, conservation of energy in the system states that

1

2mv²_o =P_BV (3.7)

wherePBV is the pressure energy. Dividing by volume on both sides of the equation gives

1

2ρv_o² =PB (3.8)

Eq. (3.8) leads to the expression for the outflow velocity vo = B_i

√µ0ρ ≡vA (3.9)

This implicates that the magnetic force accelerates the fluid to the Alfvén velocity,vA. Eq. (3.6) can be rewritten to find the velocity at which the magnetic fields reconnect,

vi= ηvo

L 1/2

= ηv²_o

Lv_o 1/2

= ηv_A²

Lv_A 1/2

= v_A

S^1/2 (3.10)

where S = Lv_A/η is the Lundquist number. It is related to the magnetic Reynolds number R_m = Lv/η, but is used when the typical plasma velocity equals the Alfvén velocity. The reconnection rate is readily written as

v_i v_A = 1

S^1/2 (3.11)

Because of the naturally occurring large Lundquist number, the Sweet-Parker reconnection rate is insufficient to explain the observed reconnection rates (Loureiro and Uzdensky, 2016). The time for magnetic energy release in a resistive current sheet (Sweet-Parker sheet) is much longer than observed. By this model, a typical solar flare should last about 2 months, while the observed duration is between 15 minutes up to

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B B

y

u

y'

x x'

u

Figure 3.2: A magnetic field line is transported by the flowu. The topology is conserved if the particles on the field line at timet₀ remains on the same line at some later time t > t0.

around 1 hour. A faster reconnection model has been proposed by Petschek (1964), who found a logarithmic dependence onS. However, Petschek’s reconnection model is still not fast enough to account for the duration of solar flares.

3.1.2 3D Reconnection

The previous section explained 2D reconnection, which occurs when E·B = 0. 2D reconnection is limited, as it only occurs at X-type null points. 3D reconnection can occur both in the presence and absence of null points, and is in general different from the 2D description. By definition, 3D reconnection occurs in regions where E·B6= 0 due to the condition

Z

E_kds6= 0 (3.12)

where the integral is along the magnetic field (Pontin, 2011). There exists a number of 3D magnetic reconnection regimes, but in this thesis we limit ourselves to strictly topological effects.

Criterion for Reconnection

Biskamp (2000) has suggested a criterion for reconnection, where he considers the local breakdown of flux conservation. This results in a change in the field line connection, and leads to a violation of the field line topology. During reconnection, particles that are originally located on a field line end up on different field lines. Biskamp’s criterion for reconnection can be explained by first taking a look at the conservation of magnetic field topology.

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Conservation of magnetic topology is associated with the conservation of magnetic flux and helicity. Flux conservation is related to Faraday’s law (2.20), where several terms are included. Magnetic topology, however, depends on the magnetic field alone. Hence conservation of magnetic topology is more generally valid than conservation of magnetic flux. Magnetic field lines can be described by a generating functionF_B(x, s, t), where FB(x, s;t) is a mapping of xalong a field line expressed in terms ofs. The parameter sis the distance along the field line. The generating function is defined by the field line equation

∂

∂sF_B=B with F_B(x,0;t) =x (3.13) A flowu(x, t) gives rise to a transport of field lines, which can be described by another generating function Fu(x, t, t0). The generating function maps the field lines at time t₀ onto field lines at time t > t₀ by the equation

∂

∂tFu=u with Fu(x, t0, t0) =x (3.14) Conservation of magnetic topology follows the same principle as the frozen-in concept, and the flow is topology conserving if the particles on a field line at time t0 remains on the same field line at some later time t > t₀. Figure 3.2 illustrates an example where two points x = FB(x,0, t0) and y = FB(y, s, t0) are located on a field line at timet0. The points are transported by the flow u to the points x⁰ = Fu(x, t, t0) and y⁰=F_u(y, t, t₀) at time t. Insertion of yinto the expression for y⁰ yields

y⁰ =F_u[F_B(x, s, t₀), t, t₀] (3.15) However, another expression fory⁰ is needed to have the topology of the magnetic field lines conserved. The condition for this is that the two points x⁰,y⁰ are connected by the same field line, as illustrated in Figure 3.2. This is the case if y⁰ = FB(x⁰, s⁰, t).

Insertion ofx⁰ into the new expression for y⁰ yields

y⁰ =F_B[F_u(x, t, t₀), s⁰, t] (3.16) A relation between Eqs. (3.15) and (3.16) is obtained, where

F_u[F_B(x, s, t₀), t, t₀] =F_B[F_u(x, t, t₀), s⁰, t] (3.17) Note that s⁰ may differ from s, and that it may be represented as a function s⁰(s, t).

Eq. (3.17) can be transformed into differential form, where the differentiation is done with respect tosandt. See Appendix A.2 for details on the calculations. Carrying out the differentiations yields the condition

∂

∂tB+u· ∇B−B· ∇u=σB (3.18)

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where σ = −_∂t^∂ _∂s^∂s⁰(s, t). Field lines can only exist if the divergence of the magnetic field vector is zero,∇·B= 0. Using this together with the vector calculus cross product identity, Eq. (3.18) can be written as

∂

∂tB− ∇ ×(u×B) =B(σ− ∇ ·u) (3.19) Faraday’s law (2.20) reveals that

E+u×B=S (3.20)

by setting

B(σ− ∇ ·u) =∇ ×S (3.21) Ideal flows are topology conserving when choosing σ = ∇ · u, hence the condition B×(∇ ×S) = 0implies field line conservation. If this condition is topology conserving, the condition

B×(∇ ×S)6= 0 (3.22)

implies a violation of the field line conservation, and is a formal criterion for reconnection (Biskamp, 2000). The vector S is deduced from Ohm’s law in Eq. (2.31). The vectorR can be split into perpendicular and parallel components, so that Ohm’s law can be rewritten as

E+v×B=R⊥+Rk (3.23)

The perpendicular component of R can be expressed as R⊥ = u⁰×B, and can be incorporated into the general velocity u in Eq. (3.20). Eq. (3.23) can then be written as

E+v×B=u⁰×B+Rk

E+ (v−u⁰)×B=R_k so that

E+u×B=Rk (3.24)

where u = v−u⁰. It can now be seen from Eq. (3.24) that if S = Rk, Eq. (3.20) is recovered. Writing the above equation into perpendicular and parallel parts, the following is seen:

E⊥+u⊥×B= 0 (3.25)

E_k =R_k (3.26)

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Hence only R_k = E_k can cause reconnection, as Eq. (3.25) is reduced to the ideal Ohm’s law. ForSto be a vector, R_k must be a vector as well. By multiplying R_k with the unit vectorbˆ =B/|B|,S becomes a vector in the direction of the magnetic field:

S=RkB/|B|

=E_kB/|B| (3.27)

Eq. (3.27) can be interpreted as a vector projection ofE onto B, so that S is parallel toB. E_k is a scalar defined by the scalar projection

E_k =|E|cosθ=E·bˆ =E· B

|B| (3.28)

UsingR instead ofE might seem more reasonable, but this will in fact end up as the same result. By taking a closer look atR, the cross productv×Bmultiplied with the unit vectorbˆ is zero! HenceR·bˆ =E·b.ˆ

The Reconnection Factor

Electrons are accelerated by currents formed under magnetic reconnection. In order to locate the acceleration sites, we need to define a variable that determines where magnetic reconnection occurs. The criterion for reconnection follows from Eq. (3.22), where the result is a new vector defined as

K=B×(∇ ×S) (3.29)

In order to locate reconnection sites, we define the reconnection factor Krec as the length ofK, so that

K_rec =|K|=|B×(∇ ×S)| (3.30)

where the updated criterion of reconnection is Krec 6= 0. Eq. (3.30) is calculated directly in Bifrost, see Section 3.3.3 for details. The value ofKrecwill vary through the simulation atmosphere as the magnetic field increases with height downwards. Because of this, we normalise all reconnection factors for each height using the Frobenius norm defined as

||A||_F = v u u t

m

X

i=1 n

X

j=1

|a_ij|²

(3.31) whereA is the array and ais each element in the array at position i, j. The norm is used as a normalising factor, which each horizontal layer is divided by. The advantage of this is that we get reconnection factors in the entire simulation atmosphere that are relatively equal in size, and comparable for all heights. We define a lower limitKmin

in order to restrain the number of reconnection sites.

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3.2 Energy Distributions 27

3.2 Energy Distributions

During magnetic reconnection, electrons from thermal plasma are accelerated to higher energies. Accelerated electrons get a new configuration where the energies become non-thermal, and they can not be described by the Maxwell-Boltzmann energy distribution. As mentioned in Section 2.3.2, electrons producing HXR bremsstrahlung are accelerated to high non-thermal energies, which can be approximated by a power-law distribution.

Coppi (1999) discuss emission from black hole accretion in terms of thermal and non- thermal particle energies, and compares the energy distributions to observations from solar system plasmas. He explains that in nature, particle distributions are probably never of one type only, and makes a concrete example by considering solar flares.

The emitting plasma from solar flares is a "hybrid" plasma, with both thermal and non-thermal components in the particle distribution. We explore the total energy distribution as a mixture of a thermal Maxwell-Boltzmann distribution with a power- law tail, in order to determine the minimum energy of the accelerated electrons.

3.2.1 Thermal Energy Distribution

Particles are in, or close to, thermal equilibrium when the temperatures are approxim- ately the same. The collisions between particles usually lead to a Maxwellian velocity distribution, with few fast particles. The thermal energy distribution of electrons is found by using the Maxwell-Boltzmann distribution for velocity and convert it into an energy distribution. The velocity distribution is written as

fMB(v)dvxdvydvz =

me

2πk_BTe

3/2

e^−m^e^v²^/2k^B^Tdvxdvydvz (3.32) where m_e and T_e is the electron mass and temperature, and k_B is the Boltzmann constant. The volume element in spherical coordinates is dvxdvydvz =v²sinθdθdφdv, so that

v²dv Z 2π

0

dφ Z π

0

sinθdθ= 4πv²dv (3.33)

Substituting (3.33) into (3.32) gives f_MB(v)dv= 4π

me

2πk_BT_e 3/2

v²e^−m^e^v²^/2k^B^Tdv (3.34) Conversion from velocity to energy is done by using that kinetic energyE= ¹₂m_ev², so that dE =mevdv = √

2meEdv. The Maxwell-Boltzmann energy distribution is then expressed as

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f_MB(E)dE= 4π

me

2πk_BT_e 3/2

2E

m_ee^−E/k^B^T^e 1

√2m_eEdE and reduced to

fMB(E) = 2

√πβ^3/2E^1/2e^−βE (3.35)

whereβ= _k¹

BTe.

3.2.2 Non-Thermal Energy Distribution

Krucker et al. (2010) suggests that the observations of HXR spectra in the corona can be explained by a single non-thermal electron population with a power-law distribution.

We assume that electrons are accelerated by magnetic reconnection in flares, and that the non-thermal energy distribution follows a power-law. The power-law distribution is expressed as

f_PL(E) =CE^−δ (3.36)

whereδ is the spectral index and C is an unknown constant. It can be argued that C must be a normalising constant in order for the power-law to be a distribution function, and making the area under the graph equal to1. The normalising integral is written as

Z ∞

E0

fPL(E)dE = Z ∞

E0

CE^−δdE= 1 (3.37)

The lower integral limit is set toE₀, which is the minimum value for which the power- law holds. The integral results in

E^1−δ 1−δ

∞

E0

= 1

C (3.38)

which in turn gives an expression for the normalising constant

C= (δ−1)E₀^δ−1 (3.39)

The power-law distribution is expressed as fPL(E) = δ−1

E0

E E0

−δ

(3.40) The power-law distribution can also be expressed in terms of the average energy from the accelerated electron beam,hEi. Instead of calculatingCas a normalising constant, it can be found using the average energy integral

Acceleration of High-Energy Electrons by Magnetic Reconnection in Coronal Nanoflares