Analysis of Tokamak plasma edge radiation with neutral uid code Eunomia

(1)

Analysis of Tokamak plasma edge radiation with

neutral fluid code Eunomia

Jaromir Camphuijsen 6042473

May 4, 2014

Verslag van Bachelorproject Natuur- en Sterrenkunde, omvang 12 EC, uitgevoerd tussen 21-01-2013 en 02-05-2014

aan de Technische Universiteit Eindhoven en FOM Instituut DIFFER ter afsluiting van de Bachelor Natuur- en Sterrenkunde

aan de FNWI van de Universiteit van Amsterdam

Supervisors TU/e: ir. Gillis Hommen dr. Marco de Baar

Supervisor UvA: dr. Aart Kleijn

(2)

Abstract

The physics of radiation in Tokamak fusion devices is explored and the pos-sible coupling of the vipos-sible Hα radiation to the Last Closed Flux Surface is studied. The radiation, caused by spontaneous emission of excited hydro-gen atoms outside the plasma, seems to be correlated to the LCFS and is used to detect the LCFS in the OFIT algorithm [1]. Using the Monte Carlo code Eunomia fed with Teand ne plasma profiles from the TCV tokamak in

Lausanne, the behavior of neutral hydrogen is simulated. This is done for a simplified model of L-mode and H-mode plasmas in a TCV like configu-ration. A radial distribution of atomic hydrogen is obtained. A CR-model code then gives the different excited states from which the Hα radiation can be calculated. The obtained radial emission profile is discussed and also a line integration of a tokamak configuration is made from an OFIT camera point of view, showing the influence of peak broadening on the camera im-ages. An interesting observation is made about the effect of the wall particle source in the simulation on the radial profiles of the L-mode plasma regime.

(3)

Samenvatting

Kernfusie is wellicht de nieuwe schone energiebron van de toekomst, maar voor het zover is moet er nog het een en ander aan de huidige fusie reactors gesleuteld worden. Om kernfusie te laten plaatsvinden moet een gas van wa-terstofatomen heel heet gemaakt worden. Dit resulteert in een plasma waar-bij de electronen en atoomkernen van elkaar ontkoppeld (ge¨ıoniseerd) zijn. Het gloeiend hete fusieplasma wordt op zijn plaats gehouden met behulp van een sterk en complex magneet veld. Om het plasma te kunnen manip-uleren met veranderingen in het magneetveld moet goed bekend zijn waar het plasma zich bevindt. Dat kan bijvoorbeeld met OFIT, een algoritme dat de locatie van de plasma rand bepaald met behulp van camerabeelden. De plasmarand blijkt namelijk straling uit te zenden afkomstig van de nog niet ge¨ıoniseerde waterstof atomen buiten het plasma die aangeslagen wor-den door de hoog energetische deeltjes uit het plasma. Dit onderzoek gaat over de relatie tussen dat uitgezonden licht en de plasmarand. De vraag waarom deze twee gekoppeld zijn wordt beantwoord en de daadwerkelijke straal afhankelijke distributie van licht emissie rond de plasmarand wordt berekend met behulp van de simulatie code Eunomia. Deze code laat in een aantal etappes een heleboel neutrale testdeeltjes uit opgegeven deeltjesbron-nen een voor een rondlopen en botsen met een opgegeven achtergrond van plasmadeeltjes en, na de eerste etappe, een zelfgegenereerde achtergrond van neutraal waterstof. Uiteindelijk komt hier een stabiele verdeling van neutraal waterstof uit. Hier kan een botsing-straling model op los gelaten worden die de aangeslagen toestanden berekent, welke weer een maat zijn voor de uit-gezonden straling. Het verkregen ´e´en dimensionale profiel laat zien dat de stralingspiek wel degelijk op of in de buurt van de plasmarand zit, maar er is wel sprake van een bepaalde spreiding. Deze spreiding zorgt op zijn beurt voor een verstoring van de verwachte piek in de camera beelden. Deze verstoring wordt berekend door coordinaten transformatie en integratie over zichtlijnen vanuit een camera standpunt. Tot slot wordt het afwijkende ra-diale profiel en de invloed van de deeltjesbron op de wand op deze afwijking bediscusieerd.

(4)

1 Introduction

Nuclear fusion is the reaction of two atom cores into one heavier core. For elements lighter than iron this reaction produces more energy than is needed to bring the two repellent nuclei together. In the sun and other stars for example, hydrogen combines into helium. As this reaction has no polluting or radioactive end products, it seems obvious to use it as a clean energy source on earth, replacing fossil fuels and nuclear fission. However because the strong coulomb force between the nuclei must be overcome, a very high temperature of approximately 10 keV (≈ 108K) is needed. This poses the problem of confinement since no known material can withstand such tem-peratures. By using the decoupling of electrons and nuclei into a plasma in which the individual particles are charged, confinement can be achieved by trapping the plasma with magnetic fields.

The Tokamak is a torus shaped device to confine and control a fusion plasma using a large amount of coils, producing a strong magnetic field. As it is the mainstream way of plasma confinement, there are many experi-mental Tokamaks around the world. In the TCV Tokamak in Switzerland, some research is focused on visible radiation from the plasma-edge, which is defined by the last closed surface of equal magnetic flux (LCFS or sep-aratrix) and separates the extremely hot ionized plasma from the colder neutral gas. The plasma-edge in Tokamak reactors seems to radiate in the visible spectrum, mainly at Hα wavelength 656.28 nm. Using optical cam-eras and the OFIT [1] detection algoritm, this radiation in turn may serve to detect the plasma-edge. Detection of the plasma edge can be used for real-time monitoring and feedback control of the magnetic confinement or subsequent scientific analysis. Plasma control is necessary to protect the inner vessel-wall material, to access specific physics regimes and optimize plasma performance.

However, to interpret the results of OFIT we need a certain understand-ing of the couplunderstand-ing of this radiation to the plasma-edge. In [1] the approxi-mation is used that the observed radiation peaks correlate exactly with the plasma edge. The purpose of this study is to test this approximation, re-fine the interpretation of the observed radiation peaks and give a detailed description of the process leading to the observed radiation. The question is whether there is a theoretical foundation for the assumption that the ob-served radiation peak lies exactly on the LCFS of the magnetic field, defining the plasma-edge. To answer this question, we would like a simulation of the radial distribution of this radiation which is produced by excited neutral hydrogen outside the plasma. We use Eunomia to perform this task, it is a Monte Carlo based code for solving the Boltzman equations, which describe neutral particle motion. Originally developed for use in linear devices like Pilot-PSI and Magnum-PSI, this is the first time it is used for simulation of a Tokamak configuration.

(6)

The first part of this thesis consists of a short study of plasma physics, the Tokamak configuration and the OFIT technique. The second part will discuss the results of the neutral particle simulations with Eunomia. Sug-gestions are made for further research on this topic.

2 Theory

In a Tokamak fusion device the production of helium from fusion of deu-terium and tritium is used:

2

1H +31H →42 He +10n (1)

The total mass on the right side of this equation is smaller by ∆m = 0.01875mp than the left side, with mp = 1.6726 × 10−27 the proton mass.

This mass difference is converted into kinetic energy of the outgoing particles according to the equation posed by Einstein:

E = ∆m × c2= 17.6MeV (2) However, to bring these particles together close enough for the reaction to occur, the coulomb force must be overcome. This is done by heating deuterium and tritium gas. As the relative speed of the particles gets bigger they can approach each other so close, the attractive strong nuclear force takes over. At these high temperatures the electrons are no longer bound to the nuclei resulting in a plasma of ionized nuclei and free electrons. Section 2.1 explores the problem of confinement of the hot fusion plasma, explaining the term LCFS and the different modes of operation. Section 2.2 and 2.3 are about the nature of the observed radiation, how it can be calculated from the simulated particle densities and a qualitative explanation why the radiation is coupled to the LCFS. Finally section 2.4 gives a short overview of the theoretical models Eunomia was based on.

2.1 Confinement

Because of the high temperatures involved (≈ 108K), the plasma in a Toka-mak cannot be confined by a physical barrier. No known material could withstand the heat. But since the gas no longer consists of neutral atoms but is now a plasma, it can be confined by the electromagnetic force. In a Tokamak reactor the plasma is confined in a toroidal shape using a com-plex magnetic field produced by several coils. The primary toroidal field is produced by external poloidal coils, it induces a plasma current. For con-finement also a poloidal field is required. This field is mainly produced by the toroidal current in the plasma itself. For plasma shaping and control, this poloidal field is complemented with toroidal currents in the primary winding and extra coils. The sum of these fields results in helical field lines

(7)

around the torus. The charged nuclei and electrons will also follow helical paths in opposite directions, while being confined by the Lorentz force. The magnetic field in combination with the current in the plasma produces a Lorentz force on the moving particles. This force is in equilibrium with the pressure of the plasma, which implies that the magnetic surfaces are also surfaces of constant pressure and that the current flows along these surfaces.

j × B = ∇p (3)

The Scrape-Off Layer is the region of the plasma where the magnetic fieldlines are not closed inside the fusion chamber but touch the reactor wall or divertor plates. Here most of the heat transport takes place due to convection rather than diffusion and the temperature gradient is very steep. The Last Closed Flux Surface (LCFS) or separatrix separates the confined plasma from the Scrape-Off Layer and is our definition of the plasma-edge. As the name suggests it is the last surface of closed magnetic flux lines.

Lack of understanding of plasma confinement has forced scientists to resort to an empirical description. We discern two main modes of confine-ment, the so called L- and H-mode. Besides internal Ohmic heating by the intrinsic plasma resistance, additional heating by RF-waves and high energy neutron beams was used in an attempt to improve confinement. However the confinement time only was found to be reduced. However, at sufficient high power input the discharge was found to make a sudden jump (the pedestal region) giving rise to twice the confinement time as before. Thus it was called the high confinement mode (H-mode), in contrast to the low confinement mode (L-mode) without pedestal. [2]

2.2 Radiation

To approximate the conditions in a fusion plasma we use the Corona model (description of the solar corona). This model assumes that all transitions to higher levels are by collision, that is: electron impact excitation, impact ionization and dielectric recombination. Also the transitions to lower levels are assumed to be radiative by nature: decay or recombination. (No photon absorption or collisional de-excitation) This model holds for low densities and high temperatures as is the case in a fusion plasma.

Because of the high energy and low density in the plasma, we can as-sume all transitions to higher energy levels are caused by electron impact. Therefore free electrons are needed to bring the hydrogen atoms in a higher state.

The radiation coming from the plasma covers a wide range of wavelengths since the electron temperature can vary from several eV outside the plasma to a keV in the plasma center and is produced by several distinct processes. Radiation in the visible region is mainly observed at the plasma edge and the Hα line clearly dominates the spectrum, together with the other Balmer

(8)

Table 1: Energy difference and wavelength as calculated from equation 4 and Einstein coefficients as calculated in [4] for spontaneous emission of the first three Balmer lines.

Name Transition ∆E(eV ) λ(nm) A (108(s−1))

Hα 3→ 2 1.89 656 0.44082910

Hβ 4→ 2 2.56 486 0.084157168 Hγ 5→ 2 2.86 434 0.025293477

lines. [3] Balmer radiation is produced when a neutral hydrogen atom falls back to the n=2 state from any higher state. The difference in energy between an electron in an atom in the n1 and the n2 quantum state and the

relation to wavelength are given by

∆En1,n2 = − µ 2¯h2 e2 4π0 " 1 n1 2 − 1 n2 2# = hc λ (4)

For hydrogen the term outside the square brackets is equal to 13.6 eV, cor-responding to the ionization energy. The probability the excited atom emits a photon by spontaneous emission is given by the so called Einstein coeffi-cients which can be calculated from the quantum mechanical description of the hydrogen atom. [4]

The energy differences, corresponding wavelengths and Einstein coeffi-cients of he first three Balmer hydrogen lines can be found in table 1. To-gether these emission lines give rise to the well known purple glow of fusion plasmas.1

2.3 LCFS radiation coupling (Qualitative)

In the region around the LCFS the neand Teprofiles show a decline because

the plasma is not confined anymore, especially steep in high confinement mode where this region is called the pedestal. Inside the separatrix the temperature rises due to high plasma density. The ingredients for Balmer radiation by collisional excitation are neutral hydrogen and high energetic collision particles which in this case will be the plasma electrons and ions. Outside the separatrix the amount of high energetic collision partners is too low to cause enough excited H atoms. Inside the separatrix, the high temperature ensures ionization of all the hydrogen so no hydrogen atoms

1

As can be seen from equation 4 the energies depend on the reduced mass µ = meM

me+M

with M the mass of the nucleus and me the mass of an electron. Because the nuclei of

deuterium and tritium contain respectively one and two neutrons, this causes a so called isotopic shift in spectral lines compared to the Balmer lines for atomic hydrogen. However the relative differences are very small, in the order of 10−3and since the only prerequisite is that the lines still fall in the visible region, this difference will not bother us.

(9)

are present to be excited. At the separatrix, ne is high enough to have

collisions, yet there is still atomic hydrogen present to get excited and fall back to the n=2 state, thereby producing the Balmer lines. This could lead to the approximate coupling of the observed visible radiation to the LCFS.

2.4 Neutral particle transport in Eunomia

The Boltzman equations describe the statistical behavior of particles in the 6 dimensional phase-space consisting of three spacial dimensions (r) and three dimensions of velocity (v) for particle species i:

∂f (r, v, i, t) ∂t = ∂f ∂t f orce + ∂f ∂t dif f usion + ∂f ∂t collision (5)

For external force F this reads:

∂ ∂t+ v · ∇r+ F_(r,v,i,t) m · ∇v f (r, v, i, t) = ∂f (r, v, i, t) ∂t collision (6)

Considering only neutral particles and neglecting gravity the external force term is lost: F = 0. In Eunomia several elastic and inelastic collisions between different neutral and plasma species (j) are considered. Therefore the collision term can be written as:

∂f (r, v, i, t) ∂t

collision

=XC(r, v, i, j) + S(r, v, i) (7) Here the first term on the right considers all elastic and inelastic momen-tum changing processes and the second term includes particle sources and sinks like gas injection and plasma recombination. Because Eunomia solves the stationary state for a certain plasma profile, there is also the condition

∂f

∂t = 0. This reduces the general Boltzmann equation 6 to:

v · ∇rf (r, v, i, t) =

X

C(r, v, i, j) + S(r, v, i) (8) [5]

3 Simulation

To examine the behaviour of the light emission at the separatrix, the dis-tribution of hydrogen atoms must be simulated. This task was performed by Eunomia, a Monte-Carlo solver for the Boltzman transport equations. It has been primarily used to simulate linear devices like Pilot-PSI and Magnum-PSI [5]. Eunomia estimates the solution to the Boltzmann equa-tion by sampling random walks of test-particles. Output from Eunomia is

(10)

then sent through a collision-radiation model which produces the final dis-tributions of different states of atomic hydrogen. Multiplied by the Einstein coefficient, this gives a quantitative model for the radial intensity of Hα radiation. To study the effect of the distribution on OFIT camera images, the radial distribution is converted to a toroidal configuration and line inte-gration is performed using Wolfram Mathematica over slines of sight (LOS) from an experimental camera perspective. Section 3.1 discusses the differ-ent geometrical represdiffer-entations that were used in the differdiffer-ent stages of the simulation. Section 3.2 and 3.3 give a short explanation of the Eunomia and CR-model code and the input that was used in this research. For a more general and detailed description of Eunomia, the CR-model and their func-tions [5] and [8] can be consulted. Section 3.4 explains further data analysis from the output of the CR-model to line integration of the LOS.

3.1 Geometry

Throughout the research different geometrical representations of the fusion plasma will be used. We can use the several symmetries of the torus to simplify the calculations. The measurements of the plasma profiles from TCV which will serve as input for Eunomia, are given in two dimensional arrays with one dimension of time and one dimension of distance normalized to the separatrixas can be seen in figure 4. As we are only interested in the radiation profile in relation to the separatrix, a simple radial profile as input for Eunomia will suffice. However Eunomia runs in two dimensions so the Eunomia geometry will consist of a rectangle with one side given by the distance from the poloidal plasma axis to the vessel wall. We discern two cases, 25 cm wall distance and 40 cm wall distance, representing two different orientations as depicted in figure 1. The other side of the rectangle is given a size of 10 cm. After the Eunomia simulation the obtained two dimensional profile will be converted to a one dimensional profile by taking the mean over this rectangle side.

The radial profiles show the spread of light emission around the sepa-ratrix, but to get an impression of the effect of the spread of the peak on the camera images a coordinate transformation was performed to account for the toroidal shape. Then the circular profile was integrated over lines of sight from a camera point of view at distance d from the secondary axis. As can be deduced from figure 2, the parametrization of the LOS reads:

x = d(1 − t) and y = d · t · tan θ (9) The x-axis corresponds to the LOS from the camera that intersects with the primary axis of the Tokamak. Considering the randomly directed pho-tons and dividing over a factor of 4πl2, where

(11)

Figure 1: An intersection of the Tokamak config-uration showing the two wall distances used to run the simulation.

Figure 2: The fusion plasma as seen from above. The parametrization of the lines of sight through the circular plasma with d the distance from the secondary axis to the camera, R the major radius, r the minor radius and θ the angle between the x-axis and the line of sight from the camera.

(12)

l = d · t

cos θ (10)

is the distance from the camera, the integrand becomes:

dI = f (p(d(1 − t))

2_{+ (d · t · tan θ)}2₎

4π(_{cos θ}d·t )2 (11)

where f (r) is the radial emission profile.

3.2 Eunomia

Eunomia simulates neutral test particles which one by one follow a random walk due to collisions with particles in the predetermined plasma background and the self generated neutral background. The geometry of the simulation has to be divided into a grid of triangle cells which can be created by the Triangle tool. Each particle will walk through the grid traveling from col-lision to colcol-lision in straight lines. Every step a test will be made if the particle exits its current cell or if the collision takes place inside the cell. In both cases the time of residence in the cell is registered. When a collision takes place, a random collision is taken from the list of possible collisions weighted by its relative probability. If it survives the collision, another step is prepared, otherwise a new particle is created. When leaving a cell, the particle checks if it hits the wall and reflects or gets absorbed, otherwise the new cell’s parameters are used to calculate when the second collision will take place. After all particles of all sources have been simulated, the neu-tral background is updated and a new simulation starts. When this neuneu-tral background converges, Eunomia is stopped. The flowchart of Eunomia is shown in figure 3.

All Eunomia input for Eunomia is given in parameter sheets. These sheets define the species of particles that will be simulated, the geometry of the vessel, the plasma profiles, all reactions that can occur between the particles, sources and sinks of particles and some other parameters to alter the simulation.

To determine which reactions will be necessary, the same databases [6] [7] as used by Eirene are consulted. Reactions are chosen to be relevant in the temperature domain of the Tokamak. The dissociation temperature of hydrogen molecules is far lower than the temperature in the Tokamak chamber so reactions concerning H2 can be ruled out. For all included

(13)

Figure 3: Flowchart of the Eunomia code [5].

Table 2: Reactions used in the Eunomia simulation. Taken from the reaction database HydHel [6] and Amjuel [7]. BGK approximation refers to the theoretical approximation.

Process Reaction Database

Elastic collision H + H → H + H BGK approximation H + p+→ H + p+ _{Amjuel 0.1T}

Impact excitation H + e−→ H ∗ +e− HydHel 2.1.1 H + p+→ H ∗ +p+ _{HydHel 3.1.2}

Ionization H + e−→ p + 2e− _{HydHel 2.1.5}

H + p+→ 2p+_{+ e}− _{HydHel 3.1.6}

Charge exchange H + p+→ p+_{+ H} _{HydHel 3.1.8}

Recombination of plasma particles into neutral hydrogen is not included in the list as this is a reaction between plasma particles and not a neutral particle reaction. The reaction was accounted for by creating a so called source of H in the source list. This recombination source is set as a volume source, it creates particles depending on the probability of recombination in a cell. A second particle source is implemented on the vessel wall, a surface source with a predetermined source strength of 1020particles per second and angular distribution, in this case a cosine distribution. Also the energy of the particles created by this source has to be set. Ionization is included in

(14)

Figure 4: Te (left) and ne (right) profiles obtained from the TCV tokamak

by the Thomson scattering method.

the reaction list and therefor forms a sink of neutral hydrogen which depends on the plasma profile. As absorption at the wall is turned on, the wall is besides a source also a sink of particles. The absorption is set to adjust its strength on the pressure at a region of 5 cm next to the wall and tries to reach an equilibrium pressure of 3.4Pa at the wall.

For the Eunomia simulation and comparison with experiment, the data from the TCV tokamak are used. The plasma profile for electron temper-ature Te and electron density ne is taken from TCV data obtained by the

Thomson-scattering method. The data for both quantities, is in the form of a two dimensional array as can be seen in figure 4, with in each cell the quantity value and a corresponding value for ρ, the minor radius normalized to the separatrix. Each row corresponds to the time of measurement and the cells on each row have different values for ρ. To obtain a one dimensional radial profile, the average was taken over time for the steady part of the profiles. In Eunomia a profile can either be a Gaussian curve or a collection of line segments. To approximate the Tokamak profiles we chose for the sec-ond option because of its simplicity and the ability to adjust certain parts of the profile. The input constructed for Eunomia is displayed in figure 5.

To account for the non symmetrical shape of the Tokamak chamber and to see how the simulation would react to different wall spacings, the wall distance was taken 25 cm for horizontal and 40 cm for vertical intersection as can be seen in figure 1. Since the original data from TCV only ranged to just outside the separatrix (≈ 23.5 cm), the profiles had to be extended for either case. The electron density had to go to zero, because outside the separatrix electrons are no longer confined. Although the temperature does not go to zero it is very low relative to the plasma temperature. It was set at a factor of 10 below the last datapoint.

(15)

Figure 5: One dimensional Te and ne profiles for a plasma in L-mode (left)

and H-mode (right). Obtained from the TCV data in figure 4 and simplified to be implemented in Eunomia. The separatrix has a typical distance of approximately 22 cm.

3.3 The CR-model

A collisional-radiative model is used to simulate the equilibrium condition of the hydrogen excited states in the neutral profile created by Eunomia. The CR-model uses its output of neutral H densities to calculate the density profiles of the first 30 excited electronic states. The model can be chosen to collect any higher excited state in the 30th state or let them result in ionization.

Table 3: Reactions producing different excited states of the hydrogen atom.

Process Reaction

Spontaneous emission H(n) → H(m < n) + hνn,m

Excitation H(n) + e− → H(m > n) + e− De-excitation H(n) + e− → H(m < n) + e−

Ionization H(n) + e− → p + 2e− Recombination p + 2e− → H(m) + e−

p + e− → H(m) + hν

The density of excited state Hn=i of atomic hydrogen is [8]

NH(n=i) = X j [Ri,j Y k(j) Nk] (12)

where Ri,j are the reaction rates of the reactions in table 3 which in turn

depend on the plasma temperature. Nk are the densities of the different

species required for each reaction j. This set of equations can then be solved for each of the excited states.

(16)

Figure 6: Radiation profile (left) and the integrated LOS (right) for discrete emission at the separatrix. The LOS with highest intensity is tangent to the separatrix. [1]

3.4 Data analysis

To convert the acquired excited state densities into photon intensities, they had to be multiplied with the corresponding Einstein coefficients An1,n2 as

given in table1.

Line integration was done over an horizontal intersection of the torus, as can be seen in figure 2. The parametrization and integration was done with Wolfram Mathematica for all four output profiles shown in figure 7. As the CR-model output data consists of discrete points, an interpolation was required. A simple linear connection between each two data points was used. This interpolated function was used as radial profile for integration over the LOS of a circular intersection of the toroidal plasma. To compare to the case of discrete emission on the separatrix, the LOS tangent to the separatrix is plotted. As can be seen in figure 6 the peak intensity should lie exactly on this line. The angle of the LOS tangent to the separatrix is given by the parametrization:

θLCF S = arcsin

0.22

d (13)

The intensity will be plotted for all θ from 1₄π to ₁₆5π.

4 Results

The densities of the hydrogen excited states calculated by the CR-model and multiplied by the Einstein coefficients found in table 1 can be found in figure 7. Each CR-model output consists of 100 data points, for the 25 cm wall spacing this means a step size of 2.5 mm and for the 40 cm case a step size of 4 mm. The first three Balmer lines are displayed since smaller wavelengths are beyond the visible region and because of low densities they are of little importance. The Hα clearly dominates through the combination of higher n=3 density and large Einstein coefficients. The total of those three is given

(17)

Table 4: Reactions producing different excited states of the hydrogen atom. Simulation Value (1022m−3s−1) Minor radius(m) Spontaneous emission 4.68365 0.224(3)

Excitation 5.46287 0.222(4)

De-excitation 7.06146 0.222(4)

Ionization 2.91579 0.226(3)

Recombination 6.4281 0.218(4)

as the solid black line, this should be closest to the observed radiation profile. The separatrix is drawn as a dashed line at 22 cm from the origin.

The results after line integration over LOS of a circular plasma with the camera at distance d = 0.6m from the toroidal axis are given in figure 8. For this integration the total radiation output profiles were used. Only the outer ring of the plasma, from the toroidal axis was calculated. Here the solid line represents at which angle θ the LOS is tangent to the separatrix.

For the deviant graph of the L-mode 40cm wall distance case, another simulation was done with a much stronger (1022 particles per second) wall source, resulting in the radial and integrated distributions in figure 9.

In table 4 the peak maximum values and their minor radius are given for all radial profiles. The error is given by the step size of the simulation.

5 Discussion

As can be seen from the data in figure 7 the emitted radiation is concentrated around the LCFS. This was already qualitatively predicted in section 2.3. However, the L mode plasma with the wall at 40cm has a second wide peak inside the plasma at about 18cm. For the H mode plasma also the 25cm case has a smaller spread opposed to the 40cm wall spacing but the 40cm cases both have a higher maximum. It should also be noted that in both 25 cm cases, the peak lies a little inside the plasma while the 40 cm has its peak almost exactly correlated with the LCFS.

After line integration in figure 8 it can be seen that the peak intensity no longer coincides with the SOL tangent to the LCFS as was found and used in [1]. All peaks are found at smaller angles than the LCFS angle, which is purely a consequence of the spread of the peak intensity in the radial profiles. For the L mode 40cm case in figure 8(b) this difference is largest as the spread of its radial profile is also very large due to the second peak.

Another simulation was done to see what caused the second peak of radi-ation inside the L-mode 40 cm plasma. So varying the wall source strength between 1020 and 1022 particles per second, it was found that the second peak inside the plasma moved towards the main peak for increasing source

(18)

(a) L mode - 25 cm (b) L mode - 40 cm

(c) H mode - 25 cm (d) H mode - 40 cm

Figure 7: The resulting output of the Eunomia/CR-model simulation mul-tiplied by the proper Einstein coefficients. Radial profiles of photon produc-tion for L and H mode plasma with separatrix on 22cm (dashed) and the wall at respectively 25cm and 40cm. The first three Balmer lines are shown, together with the total of these three (black).

(19)

(a) L mode - 25cm (b) L mode - 40cm

(c) H mode - 25cm (d) H mode - 40cm

Figure 8: Line integrated intensities for L and H mode profiles with wall distance 25 and 40cm. The horizontal axis shows the angle θ of the line of sight. The black line indicates the angle at which the SOL is tangent to the separatrix.

(20)

(a) Wall source 1020s−1- radial (b) Wall source 1022 s−1 - radial

(c) Wall source 1020 s−1 - integrated (d) Wall source 1022s−1- integrated

Figure 9: Same simulation of L-mode with wall distance 40 cm, with the wall source strength as only parameter change. On the left 1020s−1 and on the right 1022 s−1. Upper two are radial profiles and lower two integrated line intensities.

(21)

strength. In the case of a source strength of 1022 s−1, only one peak re-mains. This fact implies that there are two populations of neutral hydrogen, of which one has a very long mean free path length that extends into the plasma. These particles could possibly be recombined plasma particles with high energy which have reflected from the vessel wall. Increasing the wall source decreases the mean free path of these particles. It was also proposed that the wall source energy was very high, so the wall source would be the origin of these high energy particles. This could be true as the wall source energy is set at 2 eV, the higher source strength means more particles and so a shorter mean free path. Future simulations should be done using a lower wall source energy parameter. It is interesting to note however that only one simulation case had this second peak. H mode probably ensures ionization of even these high energy particles by the steep pedestal. Why this high energy population does not show in the 25 cm cases still has to be examined.

The plasma profiles from the TCV reactor do not extend far outside the separatrix. The extrapolations made in this thesis are reasonable and of relatively small importance but replacement by data would definitely be an improvement.

6 Conclusion

From the simulations by Eunomia some important observations can be made. First of all, the radial profile of radiation will not be discrete and this study gives a semi-quantitative result for how the profile will look like for differ-ent modes of operation and differdiffer-ent locations in the Tokamak vessel. The assumption that the peak of produced radiation is at the LCFS is approx-imately correct, however for both cases of wall distance 25 cm, the peak seems to be somewhat outside the separatrix as can be seen in figure 7 and table 4. An interesting result showed up for the L mode 40 cm wall distance profile, this is probably a consequence of wrongly chosen simulation param-eters. However, when integrated over the lines of sight from a camera point of view, the observed radiation peaks no longer lie at the same angle as the LOS tangent to the LCFS as would be the case for a discrete emission profile at the separatrix. The longer path integral for lines with smaller angles in combination with a certain spread of the profile mainly inside the plasma ensures that for line-integrated intensities now the 40 cm cases lie at smaller angles than the LOS tangent to the LCFS and the 25 cm cases of which the radial profile was shifted now correlate with the separatrix LOS. This is something that should be accounted for when bringing OFIT into practice. This effect of peak shifting by line integration changes for different orienta-tions and could contribute to deviaorienta-tions between OFIT and other separatrix detection techniques as found in [1].

(22)

Acknowledgements

I want to thank Gillis Hommen and Marco de Baar for the long waiting and good support in finishing my project and thesis. They have done a really good job at moving my not so lazy but quickly distracted and evasive ass. The whiplashes will burn for some time as a more than pleasant memory. Then there is Aart Kleijn who I want to thank for coming all the way from China and even spending some of his limited time in the Netherlands on me. Next we have some not so important people which you ought to thank in acknowledgments and makes reading them really boring. So let’s skip them and head over to one remarkable person I would like to thank: Wim Goedheer. As the name suggests, it’s a sir and he is good. He helped me for several weeks in getting to know Eunomia and solving most of the many problems I encountered. I hope he will enjoy his retirement, which I actually think he is doing because I have not been able to reach him since last summer. I imagine him constantly lying on the beach drinking coconut milk. That would actually be quite a boring thing to do after retirement, one could surely better go collecting model trains or anything of the like. Thanks to my brother for revising my English while actually being very busy in Paris doing research himself. Last (and also least) I want to thank Betsy for being my desk until she walked away and splashed mud all over my laptop.

(23)

References

[1] G. Hommen, M. de Baar, P. Nuij, G. McArdle, R. Akers, and M. Stein-buch, “Optical boundary reconstruction of tokamak plasmas for feedback control of plasma position and shape,” Review of Scientific Instruments, 2010.

[2] J. Wesson, Tokamaks. Oxford: Clarendon Press, 3 ed., 2004.

[3] O. van Hoey, “Visible light measurements on the compass tokamak,” Master’s thesis, University of Gent, 2010.

[4] Q. electrodynamics, V.B. Berestetzkii, E.M. Lifshitz, and L.P. Pitaevskii. Moscow: Nauka.

[5] R. Wieggers, B2.5-Eunomia simulations of Pilot-PSI. PhD thesis, Eind-hoven University of Technology, 2012.

[6] R. Janev, Elementary Processes in Hydrogen-Helium Plasmas - Cross Sections and Reaction Rate Coefficients. Springer-Verlag, 1987.

[7] D. Reiter, “The data file amjuel: Additional atomic and molecular data for eirene.” FZ, Forschungszentrum Jlich GmbH, 2011.

[8] S. Wolbers, “A collisional-radiative model for atomic and molecular hy-drogen in pilot-psi,” tech. rep., FOM Institute for Plasma Physics Rijn-huizen, 2011.

Analysis of Tokamak plasma edge radiation with neutral uid code Eunomia