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Aleš Podolnı́k

Study of probe diagnostics of tokamak edge plasma via computer simulation

Department of Surface and Plasma Physics

Supervisor of the doctoral thesis: doc. RNDr. Radomı́r Pánek, Ph. D.

Study programme: Physics

Study branch: Physics of Plasmas and Ionized Media

Prague 2019

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cited sources, literature and other professional sources.

I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.

In Prague, date 5^th May 2019 signature of the author

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tion

Author: Aleš Podolnı́k

Department: Department of Surface and Plasma Physics

Supervisor: doc. RNDr. Radomı́r Pánek, Ph. D., Institute of Plasma Physics of Czech Academy of Sciences

Abstract: The aim of the thesis is to examine plasma-wall interaction using computer modeling. Tokamak-relevant plasma conditions are simulated using the particle-in-cell model family SPICE working in three or two dimensions. SPICE model was upgraded with a parallel Poisson equation solver and a heat equation solver module. Plasma simulation aimed at synthetic Langmuir probe measurements were performed. First set considered a flush-mounted probe and the effect of variable magnetic field angle was studied with aim to compare existing probe data evaluation techniques and assess their operational space, in which the plasma parameters estimation via fit to the current-voltage characteristic is accurate.

Second simulation set studied a protruding probe pin. Effective collecting area of such probe was investigated with intentions of density measurement collection.

This area was found to be influenced by a combination of two factors. First, the density dampening inside the magnetic pre-sheath of the probe head, and the second, the extension of the area caused by Larmor rotation. A comparison with experimental results obtained at COMPASS tokamak was was performed, confirming these results.

Keywords: Langmuir probe, simulation, particle-in-cell, tokamak, Poisson equation, COMPASS

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Název: Studium sondových diagnostik okrajového plazmatu v tokamaku pomocı́

počı́tačových simulacı́

Autor: Aleš Podolnı́k

Katedra: Katedra fyziky povrchů a plazmatu

Vedoucı́: doc. RNDr. Radomı́r Pánek, Ph. D., Ústav fyziky plazmatu Akademie věd České republiky, v. v. i.

Abstrakt: Tato dizertačnı́ práce se zabývá studiem interakce plazmatu tokamaku se stěnou za pomocı́ počı́tačového modelovánı́, a to zejména na zkoumánı́ son- dových diagnostik. Tokamakové plazma je v nı́ simulováno pomocı́ particle-in- cell modelu SPICE pracujı́cı́ho ve dvou a třech prostorových rozměrech. V rámci práce byl model SPICE rozšı́řen o paralelnı́ výpočet Poissonovy rovnice a modul pro výpočet rovnice vedenı́tepla. Pomocı́modelu SPICE byly provedeny simulace zaměřené na napodobenı́ měřenı́ voltampérových charakteristik Langmuirových sond a to ve dvou různých geometriı́ch. Prvnı́ z nich, simulace tzv. sond zarovnaných s povrchem (flush-mounted), srovnávaly běžně použı́vané způsoby analýzy voltampérových charakteristik za účelem stanovenı́ limitů, ve kterých lze spolehlivě zı́skávat parametry plazmatu těmito metodami. Druhá sada sim- ulacı́ se zaobı́rala standardnı́m válcovým hrotem vyčnı́vajı́cı́m do plazmatu za účelem zkoumánı́ jejı́ efektivnı́ sběrné plochy, jejı́ž znalost umožňuje měřenı́ elek- tronové hustoty. Bylo zjištěno, že efektivnı́ sběrná plocha je snižována vlivem útlumu hustoty v magnetickém pre-sheathu před hlavicı́ nesoucı́ sondu a zároveň

zvyšována záchytem Larmorovsky rotujı́cı́ch částic ze vzdálenosti odpovı́dajı́cı́

jejich gyračnı́mu poloměru. Tyto výsledky byly potvrzeny srovnánı́m s měřenı́m na tokamaku COMPASS.

Klı́čová slova: Langmuirova sonda, simulace, particle-in-cell, tokamak, Poissonova rovnice, COMPASS

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selecting the ones that were the most important to me, the first is the academic environment, which should include the expert guidance of a senior scientist, a stable institutional background, as well as the inspiring discussions with colleagues in the laboratory. I was fortunate that I could join the COMPASS tokamak team and with consultations of Michael Komm and under supervision of Radomı́r Pánek, I was able to build upon foundations established by Renaud Dejarnac and Jamie Gunn. To everyone mentioned, as well as to many others, I owe my gratitude.

Second, the work is best to be ballanced with the support of the dearest ones.

I would like to thank my girlfriend Andrea, my parents and closest friends that stood by me when I needed them and were patient when I was too distracted or inconsiderate and gave me the encouragement I sometimes lacked.

And last but not least, I am grateful for the care prof. Šafránková provided to me and my fellow students, the support of doc. Kudrna, without whom this thesis would not even begin, and to my previous supervisors, doc. Plašil and prof.

Glosı́k who gave me many advice during my master’s and bachelor’s studies.

Figure 1: The contribution of a Ph. D. thesis to the total sum of human knowledge.

Adapted from [1].

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Introduction

Peaceful exploitation of nuclear fusion power has been worldwide coordinated since 1958, when world leading powers on the occasion of Fusion energy confer- ence of International Atomic Energy Agency declassified their respective research programmes [2]. Reaching to fulfil the recently discovered Lawson criterion [3], which suggested operational parameters for the reactor, multiple promising approaches were conceived shortly afterwards. They culminated with successful heating of electrons in magnetically confined plasma to 1 keV [4] in tokamak T3 at Kurchatov Institute, SSSR. Since then, the tokamak is the device that is most likely to be the foundation for first commercially available future fusion power plants. As the time progressed, multiple installations of this type were built and processes in the hot confined plasma were thoroughly studied. The pinnacle of fusion research is the International Thermonuclear Experimental Reactor (ITER), currently under construction at Cadarache, France. This project is aimed at sustaining steady state fusion reaction with the reaction energy gainQ= 10through international collaboration, applying results from various laboratories, including the Institute of Plasma Physics of the Czech Academy of Sciences (IPP CAS), either by exploiting their scientific accomplishments, or by assigning various con- tracts and tasks. At the time of writing, the first plasma discharge is planned for December 2025 [5].

Principles underlying the concept of tokamak were proposed by I. E. Tamm in 1950s [6]. The name itself is a self explanatory Russian acronym тороидальная камера с магнитными катушками (токамак), meaningtoroidal vessel with magnetic coils, now commonly written astokamak. These name-giving coils together with a current driven in the toroidally shaped plasmatic column form a magnetic field with helical field lines wrapped around the plasma. This configuration prevents the build-up of charged particles below and above the plasma column, thus suppressing drift forces and confining the ionized gas to a limited volume where the thermonuclear reactions can take place. The simplicity of the concept basis is one of the reasons why this is so far the most explored fusion plasma confinement method. Similar configuration of magnetic field can be achieved by astellarator [7], which employs meticulously calculated coil geometries to dispose of the need of the current in the plasma itself. However, the calculation of such shapes is costly, the proper design requires use of supercomputers and the coils are extremely difficult to manufacture due to precision requirements. Wendel- stein 7-X [8], largest of stellarators so far built, was commissioned only recently and still it is matched, in terms of performance, by the Joint European Torus (JET) [9], operating since 1984.

Apart from these approaches and only proven method, gravitational confinement in stars, there exist research programmes aimed at inertial confinement fusion. Formerly proposed use of nuclear weapons to heat underground water reservoirs forming heated steam was abandoned [10], and the effort has moved to studies of small deuterium-tritium filled capsules imploded by powerful lasers, such as the National Ignition Facility (NIF) [11].

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Introduction Construction of future power plants will not be possible without understand- ing of basic physical phenomena defining the behaviour of the plasma itself as well as processes taking place in various regions of the reactor. The control of core plasma properties is rather impossible without precise knowledge of the edge plasma, since these parts are closely intertwined. Mastering the edge properties enables us to operate the discharge altogether. To study the plasma itself, multiple diagnostic methods are usually employed, differing by the property or quantity they are aimed at. Most of such methods are well theoretically understood, some rely on empirical observation and scaling laws, due to enormous complexity of the system, when for example real geometry constraints prohibit the use of approxi- mations usually used in the derivation of backing theory. Some of these situations can be easily simulated by the means of computer modelling. Other contribution of the computational physics in this research include for example the study of plasma-wall interaction both regarding the plasma itself and/or its influence on immediate elements of plasma facing components (PFC), such as simulations of fuel retention or PFC heating or even melting.

This thesis is aimed at study of processes affecting ion current collection by Langmuir probe, which is a widely used diagnostic tool, via one particular method of plasma modelling, the particle-in-cell (PIC). The results of application of well- established models SPICE2 and SPICE3 will be presented as well as efforts to improving the model’s performance. Furthermore, SPICE2 has been successfully employed in simulation of heat fluxes on first wall tiles and these fluxes were used as an input of a simple heat equation solver.

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1. Nuclear fusion and fusion research devices

This chapter will describe which fusion reactions are suitable for exploitation aimed at energy production, review both historical and state-of-art devices now commonly used for fusion research. The tokamak concept will be explained in more detail, from basic parts of the device through diagnostic methods to problems that challenge the research conducted on these devices.

1.1 Nuclear fusion

Nuclear fusion is a process, in which two lighter atomic nuclei form one with larger mass. This event is either exothermic or endothermic dependent on nuclear binding energies of reactants and that of the product. As indicated in the fig. 1.1, for elements formed in the reaction producing an atom lighter than approx. 56 amu, the energy is produced, otherwise it is consumed. Exothermic properties of fusion reaction are of particular interest for low nucleon numbers A, since the energy gain per nucleon is the largest. For the further examination of fusion reaction, it is useful to describe the reaction rate of two reactants of densities n₁ and n₂ as

r =n₁n₂⟨σv⟩. (1.1)

The quantity⟨σv⟩is the mean value of the product of the reaction cross section σ and relative velocityv over Maxwellian distribution of both reacting com- pounds. The cross section σ is temperature dependent.

Figure 1.1: Binding energy per nucleon for common isotopes (adapted from [12]). Iso- topes of selected elements are highlighted. Dashed line indicates A = 56, which is in the range of most stable isotopes (⁶²Ni, ⁵⁸Fe and ⁵⁶Fe).

Fusing two nuclei together is a particularly difficult task. The nucleus is held together by the strong nuclear force, which is prominent only in extremely

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1.1. Nuclear fusion small distances compared to the nucleus size and to reach these distances, one has to overcome the repelling Coulomb force, which is considerably large, since it is proportional to 1/r². The Coulomb barrier can be overcome by quantum tunnelling, however, this effect also only manifests at small scales. It is therefore not surprising that the cross section for a fusion reaction is extremely small. To achieve considerable reaction rates, densities of reactants or their temperatures (thus relative velocities) have to be large. These conditions are easily met at high pressure high temperature environments such as star cores.

Current fusion research target is to adapt any fusion reaction as an energy source, as it has been for the stars since dawn of the universe. To do so, we have to understand the principles of the reaction as well as means of delivering the proper conditions to production.

1.1.1 Fusion in nature

As proposed by A. Eddington in [13] and later by H. Bethe [14] and numerous others, fusion reactions are main source of energy in stars. The most prominent ones are the proton-proton chain (p-p chain) and the carbon–nitrogen–oxygen cycle (CNO cycle), since they are the means of energy production in main sequence stars [15]. Both cycles involve protons to produce helium through various reaction branches.

Let’s examine most common paths of both aforementioned reactions, starting with the p-p chain. This sequence of reactions requires temperature of at least 4×10⁶K [16] to take place. It can seem that this temperature is quite low, however, one has to account for tremendous pressure inside the core. The pressure is linked with density, thus the reaction rate (1.1) increases. Nevertheless, these conditions are fulfilled even in lighter and colder stars such as the Sun.

1

1p +¹₁p−−→ ²₁D + e⁺+νe+ 0.42 MeV

2

1D +¹₁p−−→ ³₂He +γ+ 5.49 MeV

3

2He +³₂He−−→ ⁴₂He + 2¹₁p + 12.86 MeV

(1.2)

The p-p chain (1.2) [17] achieves helium production by fusing protons through

3He channel and in total produces 24.69 MeV of energy. The annihilation of two positrons emitted in the first reactions provides additional 2.04 MeV. Other branches of this chain allow production of heavier elements such as lithium, beryl- lium and boron, however, respective temperature thresholds are even larger.

For the CNO cycle, the critical temperature is approximately18×10⁶K[16].

This temperature is usually present in cores of heavier stars of the main sequence.

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The most occurrent variant is the CNO-I cycle:

1

1p +¹²₆C−−→ ¹³₇N +γ+ 1.95 MeV

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7N−−→ ¹³₆C + e⁺+ν_e+ 1.20 MeV

1

1p +¹³₆C−−→ ¹⁴₇N +γ+ 7.54 MeV

1

1p + ¹⁴₇N−−→ ¹⁵₈O +γ+ 7.35 MeV

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8O−−→ ¹⁵₇N + e⁺+ν_e+ 1.73 MeV

1

1p + ¹⁵₇N−−→ ¹²₆C +⁴₂He + 4.96 MeV

(1.3)

Total energy balance of CNO cycle is the same as of the p-p chain, even when accounted for annihilated positrons, while the former produces different numbers of gamma photons.

Stars possess several significant advantages which aid them to achieve the fusion reaction. It is the sheer size of the system and time scales that are incom- parable to human life. Both these properties make the reaction possible. It is certain that we cannot achieve both the pressure exerted by the solar mass and the extreme temperature, thus to achieve the fusion reaction, it is necessary to fulfil at least one of these conditions while keeping the other at reasonably attain- able value. Moreover, the energy density at star core is several hundreds of watts per cubic meter [15], thus it is necessary to find a reaction that is substantially more effective than simple examples presented in this section.

1.1.2 Controlled fusion

As mentioned before, achieving conditions where fusion reaction can take place is a question of reaching certain conditions. Basic principle is that the fusion reaction should run self-sufficiently, thus the power coming from the fusion should be greater than the power of losses,

P_fus > P_loss. (1.4)

It is beneficial to find such reactions that are easily achieved and produce a large amount of energy per reaction.

To describe the energy gain condition, a simple criterion can be derived. We will consider a reaction occurring in a system system with total densitynconsist- ing of two ion species A and B of equal densities n_A =n_B =n/2 and electrons.

Fusion of these ions produces a certain amount of energyE_fus per reaction. With the reaction rate r (1.1), the fusion power P_fus can be written as [3]

P_fus=rE_fus = 1

4E_fusn²⟨σv⟩ . (1.5)

Only the power loss term is left to be resolved. To do this, we define the confinement time τ_E as the ratio of total energy in the system to the power of losses:

τ_E = E_tot

P_loss. (1.6)

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1.1. Nuclear fusion It is an useful and well measurable scale of quality of the reaction system. For a system of ions and electrons of equal densitynat temperatureT, where the total energy is E_tot = 3nk_BT, the power loss term can be expressed as P_loss = ^3nk_τ^B^T (k_B is the Boltzmann constant). Energy gain condition (1.4) can be subsequentlyE

rewritten using the confinement time (1.6) and fusion power (1.5) definitions as 1

4E_fusn²⟨σv⟩ ≥ 3nk_BT τ_E , or more commonly as

nτ_E ≥ 12k_B E_fus

T

⟨σv⟩. (1.7)

The expression (1.7) is known as the Lawson criterion [3]. However, it is overly general as it does not include any particular limitations by various energy loss mechanisms or a dependence of τ_E on the temperature. To address the temperature issue, a fusion triple product criterion is thus introduced

nT τ_E ≥ 12k_B E_fus

T²

⟨σv⟩, (1.8)

which describes the system more accurately, bringing together all its substantial qualities – density, temperature and energy confinement time [18].

It has been indicated before that to construct commercially viable fusion power plant, a reaction producing maximal power should be chosen while the condition on triple product (1.8) are met. There are several such reactions, selected by the energy gain and availability of reactants, from which the optimal one can be chosen [19]:

2

1D +²₁D−−→

50 % 3

1T (1.01 MeV) +¹₁p (3.02 MeV)

−−→50 % 3

2He (0.82 MeV) +¹₀n (2.45 MeV) (1.9)

2

1D +³₁T−−→ ⁴₂He (3.5 MeV) +¹₀n (14.1 MeV) (1.10)

2

1D +³₂He−−→ ⁴₂He (3.6 MeV) +¹₁p (14.7 MeV) (1.11)

3

1T +³₁T−−→ ⁴₂He + 2¹₀n + 11.3 MeV (1.12)

3

1T +³₂He−−→

51 % 4

2He +¹₁p +¹₀n + 12.1 MeV

−−→43 % 4

2He (4.8 MeV) +²₁D (9.5 MeV)

−−→6 % 5

2He (1.89 MeV) +¹₁p (9.46 MeV)

(1.13)

In addition to conditions on triple product and energy gain, one should account for availability of required reactants. The most abundant element on Earth, with respect to these equations, is deuterium, since it is common in the water, with approx. 0.01 %[12] content amongst hydrogen atoms. Looking solely on the triple product criterion, which is plotted in fig. 1.2, the reaction (1.10) seems to be the best candidate. Deuterium-deuterium fusion (1.9) fares best in the reac- tant availability criterion, while the reaction (1.11) creates only charged products

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which can be contained using magnetic field. However, even in a simple D-D reactor, all reactions (1.9)–(1.13) would be present, since all mentioned reactants are also products of other reactions in the set. This however means that purely aneutronic fusion is almost unachievable.

Figure 1.2: Triple product for selected fusion reactions. Cross section of those reactions.

Data obtained from [19].

Looking closely on the triple product criterion at fig. 1.2, one can see that the D-T fusion reaction (1.10) is over one order of magnitude better in terms of constructing of self-sustaining reactor. However, there is only a limited supply of tritium available in the world, due to its short half-life and a process has to be devised to produce the amount of tritium needed to fuel the reactor. One possibility is a fission reaction (1.14) of lithium and neutrons [20].

1

0n +⁷₃Li−−−−−−−→

En>2 MeV”

4

2He +³₁T +¹₀n

1

0n +⁶₃Li−−−−−−−→

En<2 MeV 4

2He +³₁T (1.14)

This means that neutrons produced by D-T fusion (1.10) can be converted into tritium and thus assure the fuel production in-situ. This concept is commonly known as the tritium breeding blanket. Lithium is also fairly abundant in the Earth’s crust in amounts sufficient for sustainable energy production [21].

To pave a way for energy production from fusion reactions, the taming of the reaction of deuterium and tritium (1.10) is the most promising first step. To sus- tain the reaction, a plasma confinement technology has to be perfected and made affordable. Various auxiliary technologies such as the breeding blanket have to be developed and tested as well. The progress of past decades will soon culmi- nate in the ITER project, hopefully followed with the project of Demonstration power plant (DEMO). Following that, it is believed that the future progress in

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1.2. Fusion reactors and plasma confinement the research and development of power plants will eventually be able to devise a way to grasp the D-D fusion reaction (1.9), providing the humanity a cheap, clean and almost everlasting energy source.

1.2 Fusion reactors and plasma confinement

Energies required for D-T fusion are in such ranges that it is certain that the reacting atoms in the environment would be partially or, more probably, totally ionized. This, combined with the relatively low densities that can be achieved in terrestrial conditions, means that the fusing particles will be in the form of a plasma. The plasma can be manipulated by electric and magnetic fields, which in fact defines the means of reaction condition handling. However, this is not the only possible solution. Fusion power, although not controlled, was at first a matter of the classified weapon research, which was spectacularly made public with Ivy Mike nuclear weapon test. The mechanism is now commonly known as the inertially confined fusion (ICF). In this section, the most common experimental approaches aimed at creating a sustainable fusion reactor will be reviewed.

1.2.1 Inertial confinement

The inertial confinement devices use roughly similar approach as the internal com- bustion engines. The fusion fuel, usually in a form of a solid sphere, is brought to the reaction conditions by a compression by an external pressure source and during the process, the expanding matter and the released energy is captured by the surrounding material and subsequently converted into another form of energy.

While sounding simple when explained in this fashion, the research of this field is far from the viable solution. Two concepts of power plants driven by the inertially confined fusion reactions were pursued since 1950s. Peaceful exploitation of nuclear weaponry lead to the intention to explode full scale atomic/hydrogen bombs in underground caverns (project Pacer) suggested during the project Plowshares [22] was eventually abandoned and the inertial fusion research continues with the focus at micro-explosions.

Figure 1.3: A simple diagram of the process of target compression during the inertially confined fusion reaction.

The power plant based on ICF would explode minuscule capsules (pellets) of fusion fuel. Currently, the most promising way to induce fusion reaction is to

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heat up the outermost layer of the pellet by synchronized high-power laser beams, either directly or indirectly by heating the capsule holder (the hohlraum) first, causing an emission of X-ray radiation, which heats the capsule. The outer layer of the capsule then evaporates and becomes a plasma. The radiation pressure is however still present and together with natural expansion of the evaporated material, compresses the contents of inner layers (a simplified diagram of the process is in fig. 1.3). For application as the power plant energy source, the conditions given by the triple product criterion (1.8) should be fulfilled. At the time of writing, the most advanced experiment aimed at exploiting of the ICF is the NIF [11] which recently announced achieving of the fusion energy gain

>1[23]. This value, however, was calculated without taking the efficiency of the whole laser system into the equation. Accounting for that changes the value of the fusion gain to less than mere percent. The other issue that ICF has to find a solution for, is the conversion of the heat produced by the reaction to electrical energy.

1.2.2 Magnetic confinement

It is safe to express that the magnetically confined fusion (MCF) is currently the most advanced approach for achieving the viable production of electricity by harnessing the energy released by the fusion of deuterium and tritium atoms. The MCF relies on the simple fact that charged particles gyrate around magnetic field lines (fig. 1.4). This gyration is caused by the Lorentz force and can be described by so called cyclotron frequency [24]

ω_c= |q|B

m (1.15)

and Larmor radius

r_L = v_⊥

ω_c = mv_⊥

|q|B , (1.16)

which both describe effects of magnetic field of magnitude B acting upon a particle of mass m and velocity v⊥ perpendicular to the field direction.

Figure 1.4: An ion subjected to magnetic field gyrates around a particular field line.

The direction of rotation of electron is opposite.

Since particles follow the field lines in such arrangement and therefore they lose two degrees of freedom, a configuration of the field exploiting this fact could be favourable for confining them in a restricted volume. However, the remaining

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1.2. Fusion reactors and plasma confinement degree of freedom should be dealt with in some manner. The simplest of such arrangements is an infinitely long tube. Sadly, such a solution would require an infinite amount of fusion fuel, thus being somewhat impractical for real use.

To approach the infiniteness in closed structures, a some kind of symmetry has to be applied. One such approach exploits the effect of the magnetic mirror [25], which is an arrangement of coils aligned in such manner that the field that they produce is axially symmetric with varying intensity on the central axis. Particles entering region with higher field intensity are then repelled back to the region with lower intensity. However, for certain combinations of parallel and perpendicular (with respect to the magnetic axis) velocities, particles can exit the mirror and thus be lost, which is a problem to be solved.

Figure 1.5: A definition of directions and important quantities in the toroidal geometry.

Major (R) and minor (a) radii describe the dimensions of the torus, coordinates r (poloidal radius),θ (poloidal angle) andφ(toroidal angle) describe the position in the toroidal coordinate system. A quantity called the aspect ratio is defined as ε = a/R. Arrows describe the conventional nomenclature of directions.

Other possible arrangement is a torus (fig. 1.5). In the case of simple magnetic tube twisted around an axis perpendicular to the field direction, the particles and thus the confinement would be quickly lost, because the radial gradient of the field intensity would induce the gradB drift force and the geometry of the toroidal configuration would induce the curvature drift. Drift velocities are opposite for electrons and ions, leading to separation of the charged particles vertically. Charge separation would cause the electric field to arise and resulting E ×B drift would disrupt the plasmatic column altogether. However, if the guiding centres of particles would follow such path that the curvature andgradB drifts were compensated, no charge separation would occur and a stable plasma in the toroidal configuration could be obtained. In a simple symmetrical geometry, this can be assured when the field line that particles follow is wrapped around the plasma torus. In this arrangement, the drift direction periodically changes during the whole toroidal orbit, averaging itself to a zero value. This property of the wrapped field is described by the rotational transform [26, p. 286]

ι(r) = 2πRB_pol(r)

rB_tor(r) , (1.17)

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which is essentially defined as change of poloidal angle (see the definition of directions on the fig. 1.5) of the field line during one toroidal turn, described as the radial dependency of the ratio of poloidal magnetic field component B_pol to the toroidal component B_tor. Associated quantity is the safety factor defined as q = 2π/ι, which can be colloquially explained as number of toroidal revolutions that a particle following a field line makes during one poloidal turn. A comparison of a simple toroidal field to a field with the rotational transform is shown at the fig. 1.6. Consequently, the toroidal field with the rotational transform can be described by surfaces of constant poloidal magnetic flux ψ. Cross sections of these surfaces (in poloidal plane) can be seen on the fig. 1.5 (left). Field lines lie on these surfaces and there is one prominent amongst them, called the last closed flux surface (LCFS) (also, the separatrix). Field lines lying outside the LCFS intersect the surrounding vessel at some point, field lines lying inside the LCFS define the confined region. The volume defined by integrating a surface differential with normal parallel to the field vector along the field line is called the flux tube.

Figure 1.6: A comparison of a simple toroidal field to a field with the rotational transform (1.17). The field lines drawn on the upper picture show the outer- and innermost extent of the plasma torus, red circle depicts the LCFS at two toroidal angles3π/4apart.

A field line withq = 4on the LCFS of a torus with same dimensions (ε=a/R= 1/3) is drawn below.

How can the rotational transform be created? For example, the field with de- sired properties can be generated entirely by external coils, which is the approach the stellarators [7, 8] employ. It is incredibly difficult to compute the necessary properties of such coils for large scale machines and to manufacture them, since the precision requirements are demanding. There is, however, another possible approach. The field can be split into a poloidal and toroidal component.

The toroidal component is easy to generate with simple coils arranged around the torus. However, this construction makes the generation of the poloidal component by external coils difficult. Plasma, on the other hand, is an excellent conductor and can be used as a field-generating coil on its own. To make the plasma act

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1.2. Fusion reactors and plasma confinement as a coil, a current has to be driven in its volume. This is easy to achieve with exploiting the transformer effect. Either an iron core (as in the JET [9]) or an air core (as in the COMPASS tokamak [27]) transformer can be used. A simple explanatory sketch of the field in the tokamak is in the fig. 1.7. Using the plasma as a field generating coil can however bring some troublesome effects caused by the current-driven instabilities [28].

Figure 1.7: A sketch of the tokamak principle. A toroidal magnetic fieldis created by thetoroidal field coils. Acurrentdriven in the plasma by thecentral solenoidgenerates the poloidal magnetic field. The total field can thus be described by the rotational transform. The plasma is encased within a vacuum vessel (a section of the vessel is in the background).

The confined plasma, however, has to be limited to a finite volume. This is usually achieved either by direct physical contact of a specially designed component called the limiter, or by a particular configuration of the magnetic field that places the plasma inside the surrounding container without any immediate contact with the vessel innards. Field lines lying on the LCFS are directed to the divertor region of the tokamak. Divertor, limiter, or the vessel itself are lined with the PFCs which have to be designed to withstand the demanding conditions.

An overview drawing of the cross section of a plasma configuration in a present D-shaped tokamak can be seen on the fig. 1.8

In spite of all disadvantages of tokamaks, the underlying concept proved to be the one that sparked the interest in fusion research, when they were yet to be known. Reaching highest plasma temperatures amongst all early research devices [6], the tokamak allowed the fusion energy to be studied and fuels the hope for sustainable energy future.

Tokamak challenges

A numerous obstacles have been been discovered during the tokamak research.

While their gradual emergence and apparent inexhaustible nature seemed impossible to overcome, recent progress converges to a viable solution [5]. To operate

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Figure 1.8: Important regions of the tokamak plasma, as seen in the poloidal cross section of the divertor configuration. The flux surfaces are shaped by magnetic coils into a figure 8-like shape in the poloidal cross section. The bottom part is cut off by the divertor, leaving only the upper part for plasma to occupy. The last closed flux surface intersects the divertor on 2 concentric circles called strike lines and separates the confined plasma from the scrape-off layer. The intersections of strike lines with the poloidal plane are called the strike points. The part of the plasma in the direct contact with PFCs (and any other object immersed into the plasma) is known as the sheath.

a tokamak and to adapt it to a power plant requires to address the challenges that can be divided into several categories:

Operational challenges that affect the routine performance of the device. Up- keep of the power supplies, vacuum technology, working gas supply as well as radioactive material handling in the future, fuel cycle running including tritium breeding, neutron irradiation effects, machine component cooling and power generation etc.

Plasma confinement related challenges. Proper field generation and mitiga- tion of plasma instabilities and disruptions. Advanced confinement modes and current drive.

Plasma heating that is necessary to reach the fusion temperatures.

Plasma-surface interactions that are imminent even in case of the finest confinement performance. Heating of PFCs and fuel retention in the vessel.

Plasma diagnostics for smooth operation and feedback as well as understand- ing of processes in the core, edge and scrape-off layer (SOL) regions. Var- ious techniques are currently being used, however, only a limited subset is suitable for use in reactors.

This thesis is aimed at some of the challenges present in last two of the aforementioned categories. We will address some problems of plasma-surface interac-

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1.2. Fusion reactors and plasma confinement tions (mainly heating of a tiles subjected to high heat fluxes) and study the probe diagnostics for SOL investigation.

1.2.3 Honourable mentions

The examples provided before have stood the test of time as their success prob- ability is the most promising, nevertheless there are techniques that aim at the same goal. One of earlier attempts, which is being studied to this day, is a θ- pinch machine [29], followed by z-pinch [30] and dense plasma focus [31]. The next in the line are reversed field pinch [32], levitated dipole [33], electrostatic confinement [34], hybrid ICF and MCF concepts [35], and numerous others. It is a task for future humankind to pursue the research further and find the optimal approach.

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2. Edge plasma and probe diagnostics

As indicated in the fig. 1 in the preface, the physics presented in this thesis cover only a minuscule part of available knowledge. The same applies to the extent of the physical space, where the mechanisms studied here take part. In order to study the edge plasma and the scrape-off layer (SOL), introduction to the laws governing over the important processes and formation of dominant structures should be made. The application of these basic laws will be extended to show how the important plasma properties can be measured and what does influence these measurements.

2.1 Plasma basics

The basic definition of the plasma is that the plasma is a quasineutral gas com- prised of neutral and/or charged particles that exhibit collective behaviour. The quasineutrality condition applies on scales larger than the dimension of the plasma and the collective behaviour is governed mainly by long-range forces that are provided by the electric and the magnetic field.

Since the particles are the main object, a thorough description of their state should be given. The distribution function f(r,v) is the way to describe all necessary properties of one particle species present in the plasma. It is defined as a number of particles in the 6-dimensional phase space of velocities v and locations r:

dN =f(r,v) drdv, (2.1)

thus giving following normalization to total number of particles of the species N N =

∫︂

f(r,v) drdv, (2.2)

and more important particle density n(r) n(r) =

∫︂

f(r,v) dv. (2.3)

Similarly, any quantity α that is in any way connected with the plasma or the distribution function, can be weighted over the phase space in order to obtain its mean value:

⟨ρ⟩=

∫︂

αf(r,v) dv . (2.4)

The distribution function defined in this manner is stationary. To describe an evolution of a system, the time dependent distribution function has to be introduced. The definition of the distribution function (DF) now reads

dN(t) = f(r,v, t) drdv. (2.5)

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2.1. Plasma basics The DF is governed by the Boltzmann kinetic equation [24, p. 230]

∂f

∂t + (v · ∇_r)f + (︃F

m · ∇_v )︃

f = [︃∂f

∂t ]︃

c

, (2.6)

basically meaning that the time evolution of the distribution function is influenced by the diffusion of particles, the forces acting upon them and the collisions between the particle species (the term on the right hand side). Solving this equation gives the complete information about the system, however, it is almost impossible to solve it for other than the simplest of systems. Fortunately, there are ways to circumvent these restrictions. One of them is a simplification of the collision term [︁_∂f

∂t

]︁

c.

It can be omitted completely if the mean free path of particles the system is smaller than its characteristic length and if the collisional frequency is smaller than the plasma frequency. Generally, these conditions are met for T_e ≳ 10 eV for n₀ ≈ 1×10¹⁹m⁻³ (see the fig. 5.1 in the Chapter 5). This modification of (2.6), where, additionally, the force acting upon particles can be expressed as the Lorentz force, is known as the Vlasov equation (2.7):

∂f

∂t + (v · ∇r)f + q

m ((E +v ×B)· ∇v)f = 0, (2.7) whereE and B are the vectors of electric and magnetic field, andq is the charge of the particle species.

To address collective properties of the plasma, a fluid approach can be sufficiently accurate. To derive equations that can be used further, the moments of the Boltzmann equation have to be calculated. The first moment of the equation, given by a simple integration over the velocity space, yields theequation of continuity (2.8)

∂n

∂t +∇ ·(nu) = 0, (2.8)

which describes the evolution of plasma density with the respect to the plasma flow described by the flow velocity u = _n¹∫︁

f(r,v, t)vdv.

Multiplying the equation (2.6) by mv and integrating over the velocity space once again, the equation of motion is obtained:

nm (︃∂v

∂t + (v · ∇)v )︃

=nF − ∇ ·P+R. (2.9) This equation has introduced a pressure tensor P and the momentum change rate R given by the collisions. The force term is again often substituted with the Lorentz force giving nF = nq(E+v ×B). Accordingly, multiplying the Boltzmann equation by mv², the energy conservation law equation is given.

However, these equations apply to each of plasma particle species separately.

To obtain complete description of whole plasma, a substitution for one-fluid can be made. For a plasma consisting of electrons (indicated ase) and one ion species (indicated as i), the solution is straightforward with the addition of the Maxwell

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equations. Respective equations for each specie can be summed, the velocity re- defined as u =u_i+^m_m^e

iu_e and the current density introduced as j =en(u_i−u_e), ultimately giving the (ideal) magnetohydrodynamic (MHD) equations [18, p. 81]:

dρ

dt =−ρ·u µ₀j =∇ ×B ρdu

dt =j ×B − ∇p ∂B

∂t =−∇ ×E dp

dt =−γp∇ ·u E +u×B = 0

(2.10)

A new quantity of mass density ρ has been introduced. The last equation represents so-called ideal MHD model and should be replaced by the generalized Ohm’s law E +u×B =ηj to get the resistive MHD equations.

Equations presented in this section are the starting point for the understand- ing of basic principles. Deeper studies extend them with additional constraints and further mathematical descriptions of effects in various regions of the plasma.

The region of the interest in this thesis is the sheath and more importantly, the sheath of plasma that is present devices with magnetic field of non-negligible magnitude, which influences the distribution of particles and, subsequently, any measurements being carried out on these premises.

2.2 Plasmatic sheath

As indicated before in the fig. 1.8, the sheath is the most interesting plasma region with respect to the plasma-surface interaction. We will review its formation, characteristic quantities and mechanisms that are present in the sheath. Since in the most applications and real cases the electron sheath is present, we will further investigate a case when a conductive or a negatively biased object is in the contact with the plasma.

2.2.1 Debye shielding

To introduce the Debye shielding, and more importantly, its characteristic length, a simple plasma consisting of single charged cold ions (Z = 1) of uniform density and electrons will be considered. Electrons far from the surface are considered to be Maxwellian and the density of ions and electrons is equal there (n_i=n_e=n₀).

Generally, the potential of electric field ϕ in the plasma is given by the Poisson equation as a quantity connected to the charge density ρ:

∆ϕ=−ρ

ε₀. (2.11)

In this introduction, a 1D geometry, where∆ = _∂z^∂²2, will be studied. For a smooth transition to further chapters, the coordinate in which we are going to describe the sheath, will be designated z.

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2.2. Plasmatic sheath Near the surface, where the electric field is non-negligible, the electron density can be expressed as the Boltzmann relation

n_e=n0exp (︃ eϕ

k_BT_e )︃

, (2.12)

using the Boltzmann distribution [24, p. 9]. Expressing the total charge density asρ=e(n_i−n_e) and remembering the constant ion density, we can now rewrite the Poisson equation (2.11) to

∆ϕ=−en₀ ε0

[︃

1−exp (︃ eϕ

k_BTe

)︃]︃

. (2.13)

The exponential term can be expanded into Taylor series. Since only the first order term of the series is prevalent in the vicinity of the surface, the solution of second order differential equation is straightforward [24, p. 10].

∆ϕ =− e²n₀

ε₀k_BT_eϕ −→ ϕ(z, z ≥0) =ϕ₀exp (︃

− z λ_D

)︃

, (2.14) where ϕ₀ is the potential on the surface and λ_D is the Debye length, defined as

λ_D =

√︃ε₀k_BT_e e²n0

. (2.15)

This means that the potential in a plasma in the contact with the conductive surface decays exponentially and the characteristic length of the decay depends on the electron temperature and plasma density. For a typical tokamak SOL with parameters of T_e = 20 eV and n₀ = 1×10¹⁹m⁻³, the value of Debye length is λ_D ≈ 0.01 mm, implying the scale of any sheath-related effects. However, this value is valid for cold ion approximation in a plasma without magnetic field. Both effects will be additionally accounted for in further sections.

2.2.2 Bohm criterion

To account for the ion density, the derivation is similar. First, we define the sheath edge, with sheath edge density n_SE and potential ϕ_SE. The conditions far away from the surface remain unchanged (n_i = n_e = n_SE, ϕ = 0). The term for electron density is now rewritten using the newly defined quantities as n_e = n_SEexp(︂_e(ϕ−ϕ

SE) kBTe

)︂. To derive the criterion, we have to introduce more accurate description of ion density. We’ll consider initially cold ions coming from the bulk plasma that are further accelerated by the potential drop in the sheath.

The energy of particles has to be conserved (2.16a) as well as their number (2.16b) has to be.

1

2mv_i² =−eϕ (2.16a) n_iv_i= const. (2.16b)

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Expressing the velocity from the energy conservation law (2.16a) and substituting it to the particle conservation (2.16b), we can express the ion density in the following form:

n_i =n_SE

√︄

ϕ_SE ϕ .

The charge density that is going to define the potential through the Poisson equation is yet again the sum of ion and electron densities, rewriting the equation (2.11) to a form, where the non-uniform ion density is accounted for (compare to (2.13)):

∆ϕ =−en_SE ε₀

[︄√︄ϕ_SE

ϕ −exp (︃ eϕ

k_BT_e )︃]︄

. (2.17)

Both terms in square brackets can be expanded to Taylor series and truncated appropriately, which gives the Poisson equation the final form [36, p. 72], if we define the shifted potential as ϕ^δ=ϕ−ϕ_SE:

∆ϕ^δ ≈ en_SE ε₀

(︃ e

k_BT_e − 1 2|ϕ_SE|

)︃

ϕ^δ. (2.18)

To obtain a non-oscillating solution of the Poisson equation, the coefficient on right-hand side has to be positive, meaning that _k_B^e_T_e ≥ _2|ϕ¹

SE|. Recalling the conservation of energy (2.16a), this equation can be rewritten as m_iv²_{i, SE} ≥kBTe, which essentially means

v_{i, SE} ≥c_s ≡

√︃k_BT_e

m_i , (2.19)

i. e., the ions enter the sheath region with the speed v_{i, SE} at least equal to the sound speed. The definition of Debye length is also apparent from the equation (2.18) and is consistent with the expression (2.15) obtained before. It is evident that for cold ions, that this velocity is gained on the potential drop between bulk plasma and the sheath entrance (−ϕ_SE), in the region named as the presheath.

2.2.3 Influence of hot ions

The influence of hot ions can be accounted for both in the derivation of Debye length definition, where the ion density form is replaced by similar expression as for the Maxwellian electrons, n_i = n0exp

(︂

−_k^eϕ

BTe

)︂, and the Poisson equation is solved in the same manner. This gives the value of Debye length as:

λ_D =

√︄

ε₀k_B(T_e+T_i)

n₀e² . (2.20)

As for the Bohm criterion, the situation is more complicated. During the derivation, the same steps are performed, although the final expression is in the integral form [36, p. 76].

∫︂ ∞ 0

f_{i, SE}(v)

v² dv ≤ m_i k_BT_e

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2.2. Plasmatic sheath A simple approximation by a rectangular step function reveals that the effect of hot ions on the value of the critical speed is similar to the Debye length expression (2.20):

v_{i, SE} ≥c_{s, hot} ≡

√︄

k_B(T_e+T_i)

m_i , (2.21)

but to account for a real distribution function of a flowing plasma, the adiabatic coefficient γ should be applied to ion temperature [37], redefining the critical velocity as c_{s, hot} ≡ √︂

kB(Te+γTi)

mi . Values of γ change according to the flow type (γ = 1 for isothermal flow,γ = 5/3for adiabatic flow with isotropic pressure and γ = 3 for 1D adiabatic flow).

2.2.4 Sheath thickness

To estimate the sheath thickness, the estimation of ion current on the surface given by the Child-Langmuir law can be of use. The law is derived with the assumption that in the vicinity (z < d_s, d_s being the depletion region size) of the conductive surface, no electrons are present, since they are reflected back to the plasma, and the electron density term in the equation (2.17) is negligible. Addi- tionally, the derivative of potential at the end of the depletion region is considered to be negligible as well. However, on the contrary to the initial derivation, which was intended to determine the dependency of electrical current value between diode plates on their potential difference [38], we now apply the law to determine the sheath size using the Bohm criterion current.

The ion current density flowing onto the wall is then given by the potential of the wall (as the offset of the plasma potential of zero value) [24, p. 294]:

J_i = 4 9

√︃2e m_i

ε0|ϕ_w|³²

d²_s , (2.22)

where the sheath sized_s should now be taken as a unknown rather than a param- eter.

Since the ions surpassing the sheath edge have to fulfil the Bohm criterion, the density of the ion current flowing onto the wall is given as J_i =en₀c_s, we can calculate the sheath size d_s as

d_s =

√︄

4 9

√︃2e m_i

ε₀|ϕ_w|³² en0c_s and applying the cold ion approximation to c_s gives

d_s = 2√⁴ 2 3

⃓

⃓ eϕ_w k_BT_e

⃓

3 4

λ_D, (2.23)

which estimates the sheath size under the condition that the electron density in the sheath can be neglected and the sheath is formed by a d_s thick layer of

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positive charge. As shown in the fig. 2.2(c), this quite well estimates the region of maximal constant space charge. However, if we want to estimate the sheath boundary according to the Bohm criterion for hot ions or as a bifurcation point between ion and electron density, this is only a crude estimate. As shown in the fig. 2.2(b,e), this point is approximately twice farther in the plasma.

Figure 2.1: The sheath thickness derived from Child-Langmuir law (2.23) for deuterium plasma with electron temperature Te = 50 eV and density n0 = 1×10¹⁹m⁻³ (blue line). The value of ion current density is shown for the same parameters (red line).

A comparison of d_s and J calculated with hot ions in consideration with Ti = 2Te

is shown (dashed lines). It can be seen that the finite ion temperature increases the expected saturation current density while decreasing the sheath thickness.

The typical extent of the sheath is a few Debye lengths as can be seen in the fig. 2.1. This formula is valid for negative bias voltages and shows that the plasma influences d_s only via the electron temperature Te. Extension for hot ions is possible, substituting the value ofT_e withT_e+γT_i. The result of this extension is that the sheath thickness decreases when the ion temperature is accounted for.

Similarly, the saturation current given by the sound speed (2.21) increases.

The role of the potential atz = 0 is now clearer. The conductor is behaving as a diode according to the3/2law and more current is drawn with more negative voltage. However, increasing the voltage towards the plasma potential (to zero in this case), causes the surface to be less repulsive for electrons, eventually drawing enough current to even the ion and the electron contribution. The value of the surface potential when this occurs is known as the floating potential (ϕ_fl).

This relation defines the sheath edge as the place where the quasineutrality is already broken down and electron density exponentially reaches zero values.

This place also coincides with a point where plasma potential or electric field reach specific values. For example, Bergmann proposed the sheath edge to be at such distance, where the electric field perpendicular to the surface reaches the value of 0.1k_BT_e/λ_D [39], or the sheath edge can be defined as a distance from the conductive surface, where the ions actually reach the Bohm criterion velocity.

However, this is valid only for cold ion approximation.

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2.2. Plasmatic sheath

Figure 2.2: An overview of profiles of plasma characteristic quantities for different surface bias potentials ϕ_w. The plasma parameters are T_e = 20 eV, T_i = 40 eV, n₀ = 1×10¹⁸m⁻³ and the plasma consists of deuterium ions and electrons. Gray curves show profiles for the ϕ_w =ϕ_fl. Coloured rectangles show the extent of the sheath given by the formula (2.23). Thezaxis (perpendicular to surface) is normalized by the Debye length in the cold ion approximation.

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Investigating the real case of Debye sheath in the plasma with hot ions con- firms most of the assumptions presented here. The fig. 2.2, which has been obtained from SPICE2 simulation (with magnetic field perpendicular to the surface, which is equivalent to 1D simplification of the situation with no field) shows the profiles of important quantities in the sheath. The constants space charge edge defined by (2.23) is expanding with the applied voltage, expelling electron from locations z < d_s, showing that the assumption of negligible electron density in this area is valid. The quasineutrality is however disturbed even for z > d_s and the sheath edge is thus located farther into the plasma. Obviously, this leads to formation of a space charge and to induction of a strong electric field. Ions in the simulation are injected with velocities already distributed around c_{s, hot} (see further chapters for more on the ion injection) and are further accelerated. Total ion flux, however, remains constant, since ion density falls with z →0.

2.2.5 Influence of the magnetic field

With the introduction of magnetic field at variable inclination with respect to the surface, the situation changes substantially. Charged particles of the plasma start gyrating around field lines, stripping a degree of freedom from their movement, but due to different ion and gyration radii, they responses to electric and magnetic field in the vicinity of conductive surface differ. For simplicity, let’s assume that the B field is homogeneous and the surface is a two-dimensional plane (or, in some cases, we use 2D cross section of this geometry, usually perpendicular to the surface with magnetic field vector lying in the section plane).

Having introduced the magnetic field, a new region in the sheath appears – the magnetic presheath (also known as Chodura sheath). It starts to manifest at the Debye sheath edge and extends several Larmor radii further into the plasma.

Its characteristic length is proportional to the projection of r_L onto the z axis via the magnetic field inclination angle α_B. The total extent of the magnetic presheath is given by formula (2.24) as derived in [40].

d_mps=√ 6c_s

ω_c cosαB =√

6rLcosαB, (2.24) where the transition from _ω^c^s_c tor_Lis given by equation (1.16) as the largest Larmor radius for a particle with velocity v = c_s. For grazing angles of incidence, the angle-dependent term goes to 1 and the scale of the magnetic presheath is given by full Larmor radius. The sketch of the situation is shown in the fig. 2.3. The particles from the bulk plasma travel undisturbed along field lines towards the surface. Ions Larmor orbits are becoming deformed after they enter the magnetic presheath, while the local electric field is still too weak to substantially influence the trajectories of electrons, which are not deflected until they reach the Debye sheath boundary.

Furthermore, the size of the Debye sheath also changes. Since the density at the sheath entry changes with α_B as n_SE =n_MPEsinα_B [41], where n_MPE is the density at the magnetic presheath entry, the flux conservation ultimately gives the change in T_e, resulting in increase of the local Debye length near the sheath

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2.2. Plasmatic sheath entry, modifying the formula (2.23) to [39, 42, 43]

d_s = 0.5 λ_D

√sinα_B

⃓

⃓ eϕ_w kBTe

⃓

3 4

. (2.25)

Figure 2.3: The sheath of a plasma with external magnetic field B under an oblique angle αB (adapted from [44]). The sheath is split to two parts. The one closer to the surface is the electric sheath with the scale d_s ∼ λ_D. Farther from the surface the magnetic presheath appears, its size isd_mps∼r_L. Deeper in the plasma, the presheath is still present. Ion trajectories break from the Larmor gyration within the magnetic presheath, while electrons reach the electric sheath and are usually deflected back to the plasma after a brief period of drifting around the sheath edge.

The effect of B inclination is clearly seen on sheath profiles on the fig. 2.4, where the situation with floating surface is shown. Further into the plasma, the presheath reaches to distances similar [45] to flux tube length, since its length is given by the cross-field transport [40]. The fig. 2.4 also shows how does the extent of magnetic presheath vary with α_B. Although the profile change between 90° and 60° is marginal in absolute value, the sheath boundary d_mps suddenly increases. With α_B →0°, the location of the boundary is stabilized.

The electrostatic Debye sheath is still present, although it has been argued that for smallα_B the Debye sheath disappears. The critical field angle is derived from the formula for floating potential using a fluid model [41] and definition of µ=m_i/m_e and τ =T_i/T_e:

ϕ_fl= 1 2ln sin

[︃2π

µ (1 +τ) ]︃

(2.26) and since the condition of real result is the positive value of argument of the natural logarithm, a critical angle is found:

α^∗_B = arcsin

√︃2π

µ (1 +τ). (2.27)

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Figure 2.4: An overview of profiles of plasma characteristic quantities for different B incidence angles. The substantial change of profile shape is apparent for small angles of incidence. The sheath size d_s given by (2.25) is indicated by the dashed lines and magnetic presheath sized_mps is in dot-dashed lines limited rectangles of lighter shades of respective colors. For α_B = 90° the magnetic presheath is not present. Plasma parameters are the same as for the fig. 2.2. The axis zis perpendicular to the surface.