Spatial Structure of Limit-Cycle Oscillations


Bachelor Thesis, 2016
129 Pages, Grade: 1,3

Excerpt

Table of Contents

Preface

Abstract

Nomenclature

1 Introduction

2 Experimental setup
2.1 ASDEX Upgrade
2.1.1 Setup
2.1.2 Magnetic field
2.1.3 Divertors
2.2 Diagnostics
2.2.1 Mirnov coils
2.2.2 Bolometry
2.2.2.1 Functionality
2.2.2.2 Setup and spatial resolution

3 Theoretical background
3.1 Plasma confinement
3.1.1 Energy confinement time and Lawson criterion .
3.1.2 L-mode
3.1.3 H-mode
3.1.4 I-phase and limit-cycle oscillations
3.2 Data analysis
3.2.1 Fourier-based analysis
3.2.1.1 Fast Fourier transform
3.2.1.2 Power Spectral Density . .
3.2.1.3 Spectrogram
3.2.1.4 Example
3.2.2 Dual channel analysis
3.2.2.1 Correlation
3.2.2.2 Coherence

4 I-phase investigation
4.1 Experimental data
4.2 Identifying limit-cycle oscillations
4.2.1 Power Spectral Density
4.2.2 Spectrogram
4.3 2D-analysis of LCOs with bolometer
4.3.1 Cross-correlation
4.3.1.1 Propagation of I-phase
4.3.1.2 Spatial structure
4.3.2 Coherence
4.3.2.1 Cross Power Spectrum and Phase
4.3.2.2 Spatial structure
4.3.2.3 Comparsion with upper single null discharge .
4.3.2.4 Movement

5 Conclusion

Bibliography

List of Figures

Acknowledgments

Appendix: Source Code

Index

Nomenclature

Abbreviations

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Preface

Energy is the foundation of economic strength and technological progress. However, within this century mankind has to cope with the problem of an increasing energy requirement while fossil fuels slowly but certainly running out and people become increasingly concerned about environmental damages by current energy sources. The movement towards a rising share of renewable energy sources is a right step. However, this is linked to large financial expenditures and many of them need a lot of space (cf. solar or wind parks). Consequently, renewable energies are still a substantial minority in the energy mix. Moreover, we are hardly able to increase our energy production by another magnitude which would be helpful for another industrial revolution. Mankind has managed to become more efficient in energy utilization, however, there is still a increasing energy demand due to a constantly growing world population combined with emerging countries like China and India.

One possible solution could be nuclear fusion - bringing the process of the sun down to earth. Unfortunately, the technical implementation for an efficient power plant is relatively complicated. Scientists all over the world are researching on it since decades with the intention to fully understand the fusion plasma. One experiment is the ASDEX Upgrade of the Max Planck Institute for Plasma Physics (IPP) in Garching. Currently, the biggest experimental reactor so far - ITER- is built in southern France to prove that a nuclear fusion reactor is able to provide more energy than is necessary to start the fusion process. However, there are still many unsolved issues fusion researches have to cope with during the next decades.

This work was developed during my time as a bachelor student at the IPP and is supposed to do its bit to the success of nuclear fusion.

Zusammenfassung

In magnetisch eingeschlossenen Fusionsplasmen können grundsätzlich zwei Moden unterschieden werden: ’Low confinement’ (L-mode) und ’high confi- nement’ (H-mode). Beim Übergang zwischen diesen beiden Moden tritt eine Phase mit intermediären Einschlussbedingungen auf, die auch als I-phase be- zeichnet wird. In diesem Zwischenzustand treten Grenzzyklus-Oszillationen (LCO) im Plasma auf. Diese Oszillationen im niedrigen Kilohertz-Bereich wer- den durch Schwankungen in der Plasmatemperatur, dem Plasmadruck und weiteren wichtigen Plasmaparametern verursacht. In der vorliegenden Arbeit wird die räumliche Verteilung dieser LCOs im Plasma mit Diodenbolometern näher untersucht. Es werden Korrelations- und Kohärenz-Methoden verwen- det um die zweidimensionale Struktur der I-phase mit den Bolometer-Daten zu reproduzieren. Die Oszillationen werden in der Nähe des magnetischen X- Punktes lokalisiert. Es gibt Hinweise darauf, dass die LCOs kein stationäres Phänomen sind, sondern sich entlang der Separatrix in poloidale Richtung bewegen.

Abstract

In magnetically confined fusion plasmas it is possible to distinguish between low confinement (L-mode) and high confinement (H-mode). At the transition from L-mode to H-mode and vice versa a phase with intermediate confinement exists, which is also called I-phase. During this intermediate state limit-cycle oscillations (LCO) with a frequency in the low kilohertz range occur, corre- sponding to fluctuations in the plasma temperature, density and other key parameters. This work investigates the spatial distribution of these LCOs within the plasma by diode bolometry. The 2D spatial structure of the I- phase as it appears in the diode bolometers is resolved using correlation and coherence techniques between bolometer channels. It is found that limit-cycle oscillations are localized in the region of the magnetic X-point and it is indi- cated that they are not stationary but poloidal moving along the separatrix during the I-phase.

1 Introduction

Nuclear Fusion is an ubiquitous research topic for already more than 50 years. The idea of this research is to mimic the processes in our sun to get an almost inexhaustible energy source on earth. Our sun produces its energy mainly due to nuclear fusion of four hydrogen nuclei to one helium nucleus in three steps known as the stel- lar nucleosynthesis (figure 1.1). The mass de- fect is responsible for the release of energy. The sum of the masses of the individual hydrogen nuclei is greater than the mass of the helium nu- cleus. Corresponding to the mass-energy equiv- alence E = mc[2] from Einstein, energy is re- leased by the fusion process [13]. Consequently, it is the same reason why nuclear fission can be used to gain energy. Figure 1.2 shows the bind- ing energy per nucleon - the higher the binding energy the more stable is the nucleus. The most stable nucleus is[62]Ni [12]. Hence, the fission of heavy nuclei (e.g. uranium) releases energy as well as the fusion of light nuclei.

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Figure 1.1: Proton-proton chain: Main fusion process of the sun, that fuses four hydrogen nuclei to a he- lium nuleus in three steps. Figure adapted from [23].

In figure 1.3 we see the reaction cross section σ of several fusion processes as a function of the center-of-mass energy which corresponds to the heating power that is required in a fusion reactor. The purple line describes the last fusion process of the sun (cf. fig. 1.1). Obviously, the cross section is relatively low and the required energy is very high. However, there are other fusion processes that are more promising for the realization of nuclear fusion on earth.

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Figure 1.2: Binding energy per nucleon as a function of the mass number for different elements: Light elements release energy by fusion and heavy ones by fission. Figure adapted from [33].

The fusion of the two hydrogen isotopes Deuterium and Tritium is preferred for fu- ture fusion power plants. It has the highest cross section σ of all fusion processes shown in figure 1.3 at a comparatively low kinetic energy. The reaction equation is [29]

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whereby approximately [80] % of the released energy is carried by the neutron and can be used for further transformations to elec- tric energy using a heat exchanger to create steam, a turbine and finally a generator.

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Figure 1.3: Cross sections for some fu- sion reactions. The fusion process of the sun (purple) is very inefficient in compari- son to the preferred D-T fusion process on earth (blue). Figure adapted from [15].

In order to produce plasmas with hydrogen isotopes or helium for fusion processes on [2] earth, temperatures of roughly 10[8] K (corresponding to an energy of roughly 8.6 keV) are required. Heating the plasma to the necessary temperature as well as confining it is not a big challenge and is regularly done with two different concepts of fusion reactors - the Tokamak and the Stellarator. Both operate with a circular magnetic field. The Stellarator creates its magnetic field with a specifically shaped single coil-system, while the axisymmetric Tokamak uses the superposition of a toroidal, poloidal and vertical magnetic field. The big difference is that Stellarators are suitable for continuous operation whereas a Tokamak works in pulsed mode since a current in the plasma has to be induced by a transformer process where voltage increase is limited [18]. For future power plants it is necessary to maintain the plasma at fusion conditions for a sufficiently long time in order to be economically profitable. This task has challenged plasma physics researchers already for several decades [6, 21] in order to fully understand the plasma and how to handle it in prospect of a future power plant and still continues to do so.

In general, there are two modes of plasma confinement - the low confinement mode (L-mode) and the high confinement mode (H-mode) - and an intermediate phase in between while the transition from L- to H-mode occurs: the I-phase [4, 11]. During that phase several key plasma parameters like the temperature or density slowly change. As a consequence, plasma flows, turbulence levels and magnetic activity show variations in the low kilohertz range. This phenomenon is known as limit- cycle oscillation (LCO), however, so far it is not exactly known where in the plasma they exactly occur and if or how they propagate. The thesis at hand deals with this problem and leads to the investigation of the ’spatial structure of limit-cycle oscillations’. It will be shown that it is possible to measure the LCOs with diode bolometers and how to get a 2D spatial resolution. The aim is to localize the LCOs and to examine if and how they move within the vessel.

The following chapter 2 describes the experimental fusion reactor ASDEX Upgrade in Garching (AUG) and the diode bolometers which are the mainly utilized measuring device for this thesis. Chapter 3 provides the theoretical background to the data analysis with the priority of correlation and coherence functions. In chapter 4 the results of this method are presented by taking one plasma discharge as a prime example. Chapter 5 gives a short conclusion of the main results of this thesis and an outlook for further investigations.

2 Experimental setup

2.1 ASDEX Upgrade

2.1.1 Setup

The ASDEX Upgrade (figure 2.1) is a mid- size Tokamak fusion experiment. This re- actor type has its origins in the Soviet Union, where the first Tokamak was built in 1952 [6, 22]. AUG began operating in 1991 and is mainly used with deuterium plasmas. The minor plasma radius is 0.5 m and the major plasma radius is 1.65 m at a total volume of 14 m[3]. The total amount of plasma is 3 mg at a maximum tempera- ture of 1.5 × 10[8] K (∝ 13 keV). The total external heating capacity is about 27 MW and is composed of a neutral particle heat- ing (NBI), a ion-cyclotron-heating (ICRF) and a electron-cyclotron-heating (ECRH) system. Another component is the Ohmic heating by the electric current of up to 2 MA inside the plasma. ASDEX Upgrade distinguishes itself from other Tokamaks with its full tungsten wall [1].

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Figure 2.1: Toroidal cross section of AS- DEX Upgrade with the vacuum vessel (blue), poloidal field coils (pink) and a toroidal field coil (green). Figure adapted from [17].

For that reason it is one of the most important research experiments with respect to the upcoming experiment ITER, that is currently being built in Southern France and supposed to start operating in 2025. ITER will be four times bigger than AUG and will be the biggest experimental Tokamak reactor so far [3].

2.1 ASDEX Upgrade

2.1.2 Magnetic field

Due to the high temperature it is necessary to confine the plasma in a vacuum by a magnetic field. The required twisted magnetic field to confine the plasma in a Tokamak - indicated by the blue arrows in figure 2.2 - consists generally of three parts. The toroidal magnetic field is generated by planar copper coils (beige) and a poloidal field is formed by the current flowing in toroidal direction through the plasma (purple) which can be considered as the secondary winding of a transformer, illustrated in red. Additionally, vertical field coils, also called poloidal field coils (green), around the vessel generate a third magnetic field in vertical direction (act like Helmholtz coils) to define the plasma shape and the position of the current inside the plasma. The superposition of all magnetic fields results in a twisted field, capable of confining the plasma. This is possible, since the charged plasma particles gyrate around magnetic field lines [29]. They form so-called flux surfaces in the core of the plasma that do not have any contact to the wall up to a given minor radius. Outside this radius, defined by the so-called last closed flux surface (LCFS; green line in fig. 2.3), the field lines touch wall components in the scrape-off layer (SOL). The maximum magnetic field strength at ASDEX Upgrade is 3.9 T.

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Figure 2.2: Structure of the magnetic field of a Tokamak. It is the superposition of a toroidal (by beige coils), poloidal (by the induced plasma current IP ) and vertical field (by green coils) resulting in a twisted magnetic field (blue arrows) [10].

The limiting factor of the Tokamak is the transformer principle. To induce a direct current in the plasma the magnetic field has to change and therefore the current in the central magnetic coil (primary winding) has to increase. However, it is not possible to infinitely increase the current and consequently there is a maximum operation time for each Tokamak. The operation time of ASDEX Upgrade is limited by approximately 10 s [29].

2.1.3 Divertors

Divertors are additional components that are used to get rid of impurities in the plasma to maintain the fusion conditions which is compulsory for future long-time operations like in ITER. The above mentioned impuri- ties are the fusion product which is usually helium, but also sputtered atoms from the vessel wall. To guide the impurities to the divertors or to keep them out of the plasma the magnetic field has to be modified. There- fore, an additional coil is installed that cre- ates another magnetic field in poloidal direc- tion. Hence, the magnetic field lines in the SOL are directed to the bottom of the vac- uum vessel. Figure 2.3 shows the closed magnetic field lines (red), the LCFS, that is also called ’separatrix’(green) and the open field lines (red dashed). In general, particles in the plasma move along the field lines, how- ever, the magnetic deflection depends on the

nuclear charge. Helium has twice the charge of hydrogen and atoms from the wall are even bigger (carbon, iron, tungsten, etc.). The open field lines end up in the divertors and as a result undesirable particles get removed from the vessel by the vacuum pumps while the hydrogen plasma is kept clean. The divertors and their surroundings must be very robust since this is the only region where the plasma is in direct contact with the wall. The separatrix as well as the ’X-point’ which is the crossing point of the LCFS (cf. figure 2.3) should be kept in mind for the following investigation of the spatial structure of limit-cycle oscillations.

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Figure 2.3: Magnetic field lines with the separatrix (green). Closed flux sur- faces are illustrated by solid and open ones by dashed red lines that end up in the divertors. Figure adapted from [8].

At ASDEX Upgrade it is also possible to invert the magnetic field (upper single null configuration), implying that the X-point is placed above the plasma. For this purpose, there are also divertors on the top of the vessel. However, in this thesis the main focus lies on the usual configuration with the X-point on the bottom.

2.2 Diagnostics

2.2.1 Mirnov coils

Within the plasma, charged particles are mov- ing due to several plasma physical effects [29], meaning a current flows giving rise to mag- netic field variations. Mirnov coils are measur- ing this variation of the magnetic field, based on the integral form of Faraday’s law of induc- tion [20]

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The induced voltage is proportional to the time derivative of the magnetic field with the effec- tive area S = N · A of the coil with N windings and area A. Consequently, the voltage U at the wire ends of the coil is measured in order to observe the variation of the magnetic field.

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Figure 2.4: Positions of the 30 poloidal Mirnov coils (green) at ASDEX Upgrade. Figure adapted from [24].

At ASDEX Upgrade there are 30 Mirnov coils poloidal around the vessel shown in green in figure 2.4 and 4 coils in the toroidal plane which measure the change of the poloidal field component. However, they are not able to deliver any spatial resolution in radial direction [26]. For that reason another diagnostic method is used - bolometry.

2.2.2 Bolometry

2.2.2.1 Functionality

There are two types of bolometers at ASDEX Upgrade - foil bolometers and diode bolometers. For this work the diode bolometers, which are generally photodiodes, were the main diagnostic techniques in use. They detect electromagnetic radiation from the plasma. In the inside of the diode is a pn-junction and if the absorbed photon energy is sufficiently high (above the semiconductor bandgap), electrons get excited into the conduction band and these free carriers produce a photocurrent which is the measured signal. Depending on the current one can deduce the intensity of the absorbed radiation from the plasma on the correspondent channel. The time resolution of each camera is 5 µs, corresponding to a maximum resolvable frequency of 200 kHz which is limited by the low-pass filter [3]. That is sufficient to detect plasma phenomena in the low kilohertz range. In fact, the diodes would offer a time resolution of less than 500 ns, however, such a high sampling rate is not necessary [2]. The sensitivity range for photon detection is also quite high and ranges from ca. 1 eV to 10 keV photon energy. That corresponds to wavelengths from about 0.1 nm to 1000 nm, this means they are sensitive from the visible to the soft x-ray region and therefore called ’Absolute Extreme UltraViolet photodiodes’ (AXUV).

Unfortunately, the responsivity of these diodes is not constant but de- pends on the photon energy (see fig- ure 2.5). In particular, the low en- ergy range has a steep gradient since the AXUV are optimized to measure ultraviolet photons. For photon en- ergies higher than 200 eV the respon- sivity starts to be roughly constant [2]. However, the AXUV are reliable, have a high durability and are very robust to high temperatures and nuclear radiation.

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Figure 2.5: Photon energy dependence of the AXUV diodes. Especially for low energies the -dependence is high. The responsivity starts to be constant for energies higher than 200eV [2].

2.2.2.2 Setup and spatial resolution

ASDEX Upgrade contains eight diode bo- lometer cameras (DHC, DLX, DDC, DHS, DVC, D13, DT1, DT3) with 256 channels in total, however, only six of them with 224 channels will be used in this work since two cameras are pointing in toroidal instead of perpendicular direction into the vessel. The approximation of toroidal symmetry for Tokamaks is generally accepted, hence measuring only one cross section is suf- ficient for a representation of the whole Tokamak. Each channel measures the ra- diated energy emitted along its straight line of sight (LOS). Figure 2.6 shows the positions of the six used cameras and all their sightlines and represents that the poloidal cross section of the torus is completely covered which provides the possi- bility to get a 2D spatial resolution like a tomograph does.

For further informations on the diode char- acteristics, measurement and data acquisi- tion with the diode bolometers, see [2, 3].

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Figure 2.6: Lines of sight of all 224 used diode bolometer channels from six differ- ent modules. Modules D13, DVC, DLX, DDC measuring in the entire vessel, DHS and DLX only in the lower part. Overall, the vacuum vessel is completely covered.

3 Theoretical background

3.1 Plasma confinement

3.1.1 Energy confinement time and Lawson criterion

To achieve nuclear fusion the plasma has to be well confined. The quality of the confinement can be described with the energy confinement time τE

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where W is the total stored energy in the plasma consisting of the electron and the ion energy and P is the total (ohmic plus auxiliary, cf. sec. 2.1.1) heating power supplied to the plasma [13]. Generally, τE can be understood as the rate at which a system loses energy to its environment. In terms of nuclear fusion it is the time the plasma temperature is above the critical ignition temperature.

Furthermore, the plasma density n has to be as high as possible to increase the probability of collision and the plasma temperature T must be sufficiently high to enable the fusion particles to overcome the Coulomb barrier.

In order to achieve a self-sustained fusion plasma, the triple product of n, T and τE has to be high enough. This condition is also known as the Lawson criterion [3]

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If it is fulfilled, the plasma maintains its temperature by heating itself through the fusion reactions. Reaching this condition is absolutely necessary for future power plants in order to run them economically.

3.1.2 L-mode

In a Tokamak without external heating, the plasma is produced and heated by the ohmic transformer that simultaneously generates the plasma current. If this ohmic power is not too high, the plasma stays in the L-mode standing for ’low confinement’. In this mode the plasma density, particle confinement and temperature are relatively low and the energy transport across the separatrix as well as turbulences are quite high. In contrast, the gradients of plasma parameters at the edge are shallow [3].

3.1.3 H-mode

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Figure 3.1: Pressure profiles of L- mode and H-mode in comparison. The profile of the H-mode seems to be put on a pedestal causing a steep gradient at the edge where a transport barrier appears slowing down the flow of heat energy out of the plasma. r/a = 1 denotes the plasma boundary. Figure adapted from [38].

The transition from L-mode to H-mode is a quite spontaneous process that starts occur- ring when the significant threshold value for the heating power is researched, that depends on the size of the machine and further pa- rameters. Initially, the H-mode was found by auxiliary heating with NBI [35]. The H stands for ’high confinement’, meaning that the energy confinement time τE is enhanced by a factor of approximately two. This is the most important aspect of an H-mode plasma compared with an L-mode plasma [30]. Fig- ure 3.1 shows the pressure profiles of the L- mode and H-mode in comparison. The H- mode is distinguished by an edge pedestal in the profiles, meaning that the energy trans- port across the separatrix is clearly lower than in the L-mode (transport barrier) [13] and the central plasma pressure is increased. This is caused by steep gradients of the elec- tron and ion density as well as the temperature inside the separatrix, that are based on spontaneously generated sheared plasma flows (zonal flows) at the edge, which suppress turbulence in the edge plasma by taking energy from them [3, 5, 34]. The profile gradients in the core stay the same, consequently, it looks like putting the L-mode profiles on a pedestal (cf. figure 3.1) [30]. Nevertheless, there is a density limit for plasmas in Tokamaks, known as the Greenwald limit [16]. Getting closer to this limit the plasma confinement decreases rapidly [3]. Furthermore, the steep gradients at the edge of the plasma result in instabilities during the H-mode known as ’Edge Localised Modes’ (type-I ELMs). This is caused by a periodical relaxation of the plasma pressure profile to a less steep gradient. After an ELM the profile steepens again until another ELM occurs. During an ELM many particles and a significant amount of heat escapes from the plasma leading to strong stress on the vacuum chamber components [37]. However, the H-mode - which was discovered in the eighties on the ASDEX machine [35] - is still considered as the most suitable mode for future fusion reactors respectively experiments like ITER or DEMO [28].

3.1.4 I-phase and limit-cycle oscillations

The period between L- and H-mode is often referred to as I-phase. This intermediate phase is not yet fully understood, however, the assumption is that a nonlinear process causes the transition. The turbulence suppression at the L-H transition is generated by a strong and localized radial electric field giving rise to strong shear flows in the transport barrier (cf. fig. 3.1) [19].

The I-phase is characterized by limit-cycle oscillations illustrated in figure 3.2. It depicts a transition from a L-mode to a H-mode with a clear I-phase in between. The left-hand side shows the entire transition initiated by the auxiliary heating Ptot (a). Plasma density (b), temperature (c) and divertor current (d) start to increase immediately until reaching the H-mode that is recognizable due to the occurrence of type-I ELMs. During the I-phase (right-hand side) oscillations in many plasma quantities like plasma velocity (e), divertor current (g), plasma density (i), etc. occur with the LCO-frequency which in the vast majority of cases lies within the range between 0.5 kHz and 5 kHz depending on the edge density ne,edge and electron temperature Te.

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Figure 3.2: L- to H-Mode transition with a clearly visible I-phase in between. The transition was triggerd by switching on the auxiliary heating power (a). In consequence, the density (b) and temperature (c) increased. The H-mode is identified by the appearance of type-I-ELMs (d). Limit-cycle oscillations during the I-phase are visible in several key plasma parameters (right-hand half) like the plasma velocity (e), divertor current (g) and the plasma density (i) [4].

Usually, the I-phase only takes place for a few microseconds, but if the parameter for heating the plasma and density are chosen right it could last for some seconds in ASDEX Upgrade, where the maximum plasma duration is 10 s. In fact, it is possible to maintain the I-phase for the whole plasma discharge if the parameters stay near the transition threshold [11]. It has been shown, that it is possible to reach an H-mode without any type-I ELMs after staying longer in the I-phase [4]. The transition from the L-mode to the I-phase is rather sharp, but the transition into the H-mode is not that obvious, it evolves softly from the I-phase and is often only recognized by type-I ELMs [11].

It is still not possible to say, if the LCO cause the oscillations in the radial electric field or vice versa (both directions have been reported [9, 27]), but obviously both phenomena happen during the I-phase [19].

However, the frequency scaling has already been investigated with the Mirnov coils where a heuristic formula was found to assess the LCO-frequency

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with the density on the plasma edgene,edge, the electron temperature Te, the toroidal magnetic field strength Bt and the edge safety factor q95 [4].

3.2 Data analysis

3.2.1 Fourier-based analysis

3.2.1.1 Fast Fourier transform

The Fourier transform is used to get all the frequencies with their amplitudes that a given signal contains. The ordinary Fourier transform F (ω) of a signal f (t) in one dimension is known as

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In general, it takes a periodic signal in the time domain and transforms it into the frequency domain. In practice a continuously measured signal does not exist, a measurement usually consists of discrete data points. Moreover, real signals are neither periodic nor infinite. The solution to get the Fourier transform of a real signal anyway, is the Discrete Fourier transform (DFT). Essentially, the integral in equation 3.4 is superseded by a sum,

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with the sampling period Ts and the number of samples N . Using the total length of the signal L, the sampling period is calculated as Ts = L/N . In order to resolve high frequencies, the sampling period Ts has to be small enough respectively the number of samples N big enough for a given signal length.

The highest resolvable frequency fmax (Nyquist-frequency) is determined by the Nyquist-Shannon sampling theorem considering the sampling frequency fs and states

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In principle it states that one has to sample the signal at least by the twofold of the maximum frequency fmax which is desired to be resolved.

The Fast Fourier transform (FFT) is a more efficient method to calculate the DFT by using symmetries in the calculated terms. For the DFT the number of calculations scales with 2N[2] and for the the FFT with 2N log N [7, 36]. Thus, already for N = 1024 the FFT is more than 99% faster compared to the DFT.

3.2.1.2 Power Spectral Density

Based on the FFT, the Power Spectral Density (PSD) provides the distribution of a signal into its frequency components with the power of each frequency within the signal. Hence, it is also known as power spectrum

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where x(ω) is the Fourier transform of a segment of the signal. The whole signal gets divided into a distinct number of segments with same length and the FFT of each segment will be calculated. Then the square of the absolute value |x∗ · x| for each segment is calculated. If the signal was complex, also the Fourier transform would be complex and the complex-conjugated value x∗(ω) would have to be taken. Finally, the average over all segments is calculated (< [...] >).

In the spectrum of the FFT - without averaging over sub-segments - every frequency contained in the signal is shown with its amplitude, regardless if it exists over the whole time or only for a few oscillations. The advantage of the PSD is, that by averaging frequencies that exist over the whole signal time get enhanced and others get suppressed. Instead of only using the FFT-spectrum, this way it is much easier to identify such consistent frequencies although their amplitude is probably relatively low. Consequently, the PSD is more helpful to identify LCO-frequencies, if they only generate a very low signal on some channels.

3.2.1.3 Spectrogram

The spectrogram is a two dimensional function, depending on time and frequency. Like the FFT, the spectrogram splits the signal into its frequency components, but in a time dependent depiction. For each time interval he spectrogram shows the frequencies appearing in the signal. Usually, the illustration is a contour plot. The time is plotted on the x-axis and frequencies on the y-axis, while the corresponding frequency share for each frequency at each time is plotted as a third dimension with a colorbar. This frequency share is interpreted as the absolute square value of the Fourier transform. Actually, it is the Short-time Fourier transform (STFT), meaning that the signal is splitted into small consecutive segments which possibly overlap while the FFT for each segment is calculated [14].

In general, the time and frequency resolution of the spectrograms is determined by the length of the segments and the number of FFT points (NFFT) that are used to calculate the STFT in each segment. The greater NFFT the longer the total calculation takes. If NFFT is greater than the number of data points in the segment, the data will be zero-padded to NFFT. Obviously, a narrow segment results in a fine time resolution but a coarse frequency resolution and vice versa. Hence, in the case of a spectrogram there is a trade-off between time and frequency resolution.

3.2.1.4 Example

To illustrate the Fourier analysis an example will be considered. The test signal is the function

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with A1 = 3.0, A2 = 2.0, A3 = 5.0, f1 = 3.0 Hz, f2 = 20.0 Hz and f3(t) increasing over time from 1.0 Hz to 50.0 Hz in 1000 equidistant steps. In this example the total time is 4s, which is divided into 1000 intervals. Thus, the time step is dt = 4 ms, corresponding to a sampling frequency of 250 Hz and by taking equation 3.6 into consideration the maximum resolvable frequency is 125 Hz.

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Figure 3.3: Example to illustrate Fourier-based analysis: (a) the given signal s(t), (b) FFT, (c) PSD and (d) the spectrogram. The three different frequency components f1, f2 and f3(t) are clearly visible. f1 and f2 cause the peaks in the FFT and PSD and f3(t) is the background. The spectrogram shows the time dependence of the frequencies.

Figure 3.3(a) shows s(t) with the clearly visible increasing frequency f3. The con- tinuous prevailing frequencies 3.0 Hz and 20.0 Hz are easy to identify in the FFT (b), PSD (c) and the spectrogram (d). In the PSD the peaks are broader as a con- sequence of having smaller time intervals (Δf = 1/ΔT ) due to dividing the signal in segments as described above (sec. 3.2.1.2). The advantage of the PSD becomes more obvious later on by taking much more frequencies into account. The time dependent frequency already provides some noise in the PSD and FFT. However, in the spectrogram the 3.0 Hz and 20.0 Hz are still clearly visible as well as the time dependent frequency f3.

3.2.2 Dual channel analysis

The aim of this work is to get the spatial structure of limit-cycle oscillations by using 224 bolometer channels described in section 2.2.2.2. Each channel provides a signal measured along its line of sight. However, at least two channels - that have an intersection within the cross section of the plasma vessel - are necessary to localize the origin of the signal in two dimensions. In order to decide if both channels are measuring the same signal, one has to use mathematical methods like correlation and coherence analysis.

3.2.2.1 Correlation

The correlation function tells how similar two signals are as a function of the time displacement τ between both signals. The value is always between -1 (total anticorrelated) and 1 (total correlated) if the correlation is normalized to the standard deviation of the input signals.

The correlation between two standardized functions [illustration not visible in this excerpt] and [illustration not visible in this excerpt] is defined as

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The larger Rxy becomes the higher the correlation is. The position of the maximum of Rxy provides the time shift τ between both signals. Usually, the cross-correlation is calculated for two different functions x and y. If they are the same function this is also known as the auto-correlation. The Fourier transform of the auto-correlation

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results in the PSD (eq. 3.7) after averaging [illustration not visible in this excerpt]. This is also known as the Wiener-Khinchin theorem [25].

Figure 3.4: Two different functions x and y, which are chosen to explain the cross- correlation in a qualitative manner. Later on, the same will be done with bolometer signals.

Let us consider two different functions x(t) and y(t) shown in the following figure 3.4 to illustrate the calculation of the cross-correlation function which is depicted in figure 3.5.

The first step is to shift the second function y by a time displacement τ1 (fig. 3.5 b). In this case it is shifted to the right with τ1 resulting in y(t − τ1). Then, the product between x and y(t − τ1) is calculated (fig. 3.5 c). This prod- uct function gets integrated over time with the result of the correlation value for time lag τ1 correlation in a qualitative manner. Later on, the same will be done with bolome (fig. 3.5 d). The entire cross-correlation function (fig. 3.5 e) follows by doing this for every possible time shift τi.

If two signals have the same periodicity - what we expect due to measuring the LCO-frequencies - only shifted in phase, obviously, the correlation function will be periodic, too.

Cross-correlation is closely related to the convolution

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Cross-correlation is obtained from convolution by mirroring the first function, then calculat- ing the convolution with the second signal and finally substitute the time variable t with the time shift variable τ . They are the same if one function is symmetric [32].

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Figure 3.5: Cross-correlation for the functions x and y: At first, func- tion y is shifted in time by τ1 (b), then the product with x is calcu- lated (c) and this product function gets integrated (d). Finally, these steps are done for all possible time shifts τi to get the total correlation- function (e).

3.2.2.2 Coherence

Correlation exhibits how similar two signals are, however, it is not clear at which frequencies the signals are correlated. For that reason another mathematical method is required in comparing signals to find out at which frequencies signals are correlated. This is provided by coherence analysis.

At first, the Cross Power Spectrum (CPS; also ’cross-spectral density’)

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has to be calculated. It is quite similar with the PSD (formula 3.7), but taking two different functions. Analogous to relation 3.10, the CPS and the cross-correlation Sxy>)

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Coherence Kxy is the square of the absolute value of the CPS Sxy , normalized with the PSDs Sxx and Syy of both functions

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It is a measure for the similarity of two signals as a function of frequency. The value of Kxy is always real and between 0 and 1.

In order to know the lead or lag of one signal to the other, one has to calculate the phase shift, wherefore it is necessary to determine the real and imaginary part of the CPS. There are two ways to implement this.

One approach is using relation 3.13 to rewrite the CPS as a product of an amplitude function Axy (ω) (’cross-amplitude spectrum’) and a phase function Φxy (ω) = Φy −Φx (’phase spectrum’) [31]:

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Obviously, Φxy (ω) provides the phase shift between both signals for all frequencies including the LCO-frequency.

The CPS can also be written in polar coordinates as [25]

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with the ’coincident spectra’ Cxy (real part of Sxy ; even function) and the ’quadrature spectra’ (imaginary part; odd function). Both describe the degree of co-oscillation for the frequency constituents of the two signals, but the first is for the fluctuations in phase (phase difference 0 or π) and the second for those out of phase by ±π/2 [31]. As for every complex number the phase in this case is

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4 I-phase investigation

4.1 Experimental data

The data used in this work originates from the diode bolometers explained in section 2.2.2. They deliver a signal that is proportional to the amplitude of the absorbed electromagnetic radiation with a time resolution of 2 µs corresponding to a sam- pling frequency of 500 kHz. This resolution is absolutely sufficient to see the LCO- frequencies applying Fourier based analysis since the Nyquist frequency is 250 kHz (cf. section 3.2.1.1). Taking the entire signal of a discharge of approximately 10 s one would have a data-array with 5 million entries. To shorten the following calcu- lations it is strongly recommended to take as few data as possible, but as many as necessary. To get at least 50 peaks of the limit-cycle oscillations, also at the lowest estimated LCO-frequency of 0.5 kHz it is sufficient to take an extract of 100 ms. One could choose a longer period to get more reliable results, however, it is endorsed to use not more than one second, otherwise the following calculations take up too much computing time. Moreover, a lot of disk space is used if the signals are saved. One data array of 0.5 s has about 2.5 MB and therefore only raw data files with a total size of about 1 GB (≈ 2 · 224 · 2.5 MB) are used.

4.2 Identifying limit-cycle oscillations

4.2.1 Power Spectral Density

The Power Spectral Density is the most convenient method to identify limit-cycle oscillations and their characteristic frequency. Even higher harmonics are possibly visible. Plasma discharge #31494 is excellently suited to examine the LCOs due to its long I-phase of approximately 0.5 s and an almost constant LCO-frequency. Obviously, in this discharge most power is contained in the constant LCO-frequency which is the most dominant peak in the PSD (figure 4.2). For this diagram channel 13 from module DHC pointing in the lower half of the vessel shown in figure 4.1 was taken as reference channel.

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Figure 4.1: LOS of DHC channel 13 and location of the module as well as of Mirnov coil MHE-C09-23.

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Figure 4.2: PSD of discharge #31494. The LCO-frequency and even higher harmonics of it are clearly visible.

In such cases, the LCO-frequency can also clearly be seen in the absolute value of the FFT, however, considering another discharge with a shorter I-phase and a non- constant LCO-frequency (#29306) shows the advantage of the PSD (figure 4.3). There, the LCO-frequency is not constant but decreasing (see figure 4.4), thus, significantly more frequencies appear. The LCO-frequency is no longer visible in the absolute value of the FFT (fig. 4.3(a)), though, in the PSD (fig. 4.3(b)) as a broader peak around 1 kHz which is considerably separated from the background. Basically, for further investigations it is useful to consider a time frame where the LCO-frequency is as constant as possible.

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(a) FFT (b) PSD

Figure 4.3: Discharge #29306 illustrates the advantage of the PSD towards FFT. The LCO-frequency is still visible in the PSD while it is no longer identifiable in the FFT.

4.2.2 Spectrogram

In the example of a spectrogram in section 3.2.1.4 all three frequencies are clearly visible, although superpositions occur. However, in practice there are much more frequencies in the measured bolometer signal and it is possible that parasitic fre- quencies provide such a strong signal that the constant frequencies - in practice the LCO-frequency - are not visible. Another problem is, that the signal of the diode bolometer is proportional to the emitted power of the plasma. Thus, on the bottom of the vessel where the divertors are, the signal is always stronger. Hence, with the real signal data, it is necessary to choose an intensity scale for the spectrogram in such a way that the LCO-frequencies are always visible and in an appropriate contrast to all the other parasitic frequencies from the plasma.

Furthermore, the responsivity of the AXUV diodes according to the photon energy is not constant as illustrated in section 2.2.2.1. Especially in the low energy spectrum, which is typical for investigating the limit-cycle oscillations, a steep gradient occurs.

In previous works [4], the LCOs have already been detected with the Mirnov coils (section 2.2.1). During the limit-cycle oscillations the magnetic field changes signifi- cantly, therefore, Mirnov coils (cf. section 2.2.1) are very suitable to identify LCOs. To be sure that the bolometers measure the same LCO-frequency with the identi- cal time resolution as the Mirnov coils the spectrograms of discharge #29306 and #31494 are compared in figure 4.4 and 4.5. Therefore, diode bolometer ’DHC-13’ and Mirnov coil ’MHE-C09-23’, shown in figure 4.1, were used since both of them are measuring in the lower area of the vacuum vessel where the signals are stronger. As described in section 3.2.1.3 the time and frequency resolution of the spectrograms depends on the value of NFFT. In the following cases 64 segments and NFFT=4096 were choosen for the bolometer data. The 64 time segments on the x-axis are clearly visible as well as the 41 frequency windows of approximately Δf = 122 Hz on the y-axis. Nevertheless, the similarity to the spectrogram of the magnetic data, were other parameters were chosen, is striking. Figure 4.4 shows die comparison for plasma discharge #29306 with a decreasing LCO-frequency and figure 4.5 for discharge #31494 with the constant LCO-frequency.

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Figure 4.4: Spectrograms of discharge #29306 in comparison for magnetic (a) and bolometer (b) signals. Obviously, they are very similar in time and frequency domain. Hence, it is justifiable to use the diode bolometers for further investigations.

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Figure 4.5: Spectrograms of discharge #31494 in comparison for magnetic (a) and bolometer (b) signals.

4.3 2D-analysis of LCOs with bolometer

4.3.1 Cross-correlation

4.3.1.1 Propagation of I-phase

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Figure 4.6: All 48 DHC-channels of the bolometer diagnostics. The red LOS (DHC-13) is the reference channel for the propagation analysis. Channels are counted from the bottom (1) to the top (48).

The cross-correlation function is a measure of similarity of two signals depending on their time displacement τ as described in 3.2.2.1. Comparing all the channels within one camera enables to get the development of the time displacement relative to one reference channel, that corresponds to the movement of the signal. The data of cam- era ’DHC’ from plasma discharge #31494 are used with channel 13 as the reference channel. Figure 4.6 shows all the channels of the mentioned camera, that are num- bered from the bottom to the top. 48 cor- relation functions will be obtained. An ex- ample is shown in figure 4.7 displaying the correlation of channel DHC-5 with the ref- erence channel DHC-13. Since both signals contain a periodic oscillation, the cor- relation function is periodic, too (cf. sec- tion 3.2.2.1). The maximum correlation value (MCV) is slightly shifted in time and amounts to τ = 160 µs in this example. Plotting all these time shifts from the 48 correlation functions over the channel number, one obtains the graph in figure 4.8. It is evident that the channel 8 till 12 have a negative time lag and the channel from 14 to 48 overall a positive one compared to channel 13. The strong decrease from channel 36 to 37 is due to a phase shift. This means, that the signals on the lower channels are earlier measured than on the upper channels indicating that the LCOs are moving from bottom to top. This result has already been seen with the data from the Mirnov coils [4], however, there was a maximum time lag from the bottom to the top of τmax = 124 µs found whereas the bolometer data show a value that is almost twice as high. It seems to be that the propagation velocity that was calculated to vpulse ≈ 91 km s−[1] in [4] is not constant but depends on other parameters, probably the LCO-frequency itself. This value was determined with discharge #29302 where die LCO-frequency for the considered time frame is approximately 2.5 kHz in contrast to a constant LCO-frequency of ≈1.6 kHz for discharge #31494 (cf. fig. 4.5).

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Figure 4.7: Cross-correlation function of the channels DHC-13 and DHC-5 for #31494 with time lag τ ≈ 0.16 ms and a maximum correlation value of ≈0.7.

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Figure 4.8: Time lag between reference chan- nel DHC-13 and all the other channels of mod- ule DHC as a function of the channel number (1-48). An increasing time lag from Channel 8 to 44 is visible with a phase shift between channel 36 and 37.

It is not always possible to get as clear results for the correlation function as in figure 4.7 and the time development as a function of the channel number as in figure 4.8. Generally, two problems - shown in figures 4.9 and 4.10 - occur in practice.

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Figure 4.9: The two channels are hardly correlated at the LCO-frequency. There- fore the correlation function is domi- nated by other signals and the time shift at the maximum correlation value corre- sponds not to the LCO-frequency.

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Figure 4.10: Cross-correlation function disturbed by other frequencies. The LCO-frequency provides the sinusoidal envelope function but the delivered time shift at the MCV is wrong. The δ-Peak at τ =0 indicates coincidental noise.

On the one hand it is possible that the amplitude at the LCO-frequency is very low in one channel. Consequently, the correlation value in total is relatively low and probably dominated by another frequency with a different time displacement that is often too big. This problem is illustrated in figure 4.9. To avoid using these values that are apparently not linked to the LCOs, a threshold of ±250 µs is chosen that is responsible for the gaps in figure 4.8.

On the other hand both channels are probably correlated at the LCO-frequency, but also strongly correlated at some other frequencies. The result is shown in figure 4.10. Although using a smoothing function before calculating the cross-correlation function the higher frequencies are still clearly visible. The sinusoidal envelope function is determined by the LCO-frequency (approximately 1.6 kHz). Obviously, the peak of the sinusoidal envelope function is shifted to a positive time displacement of approx. τ = 75 µs, however, the MCV is found at τ = 0. In that case, this δ-Peak indicates that both channels are correlated in coincidental noise, too, because the correlation function of coincidental noise is such a δ-Peak at τ = 0.

4.3.1.2 Spatial structure

The indication that the LCOs are moving up- wards does not provide any information about the radial position. In order to localize the LCOs in the 2D cross section it is indispens- able to use the intersection points of all 224 LOS. In a first step all the channels with a significant amplitude in the LCO-frequency must be selected. This is done by the identi- fication in the spectrogram (see section 4.2.2) and manual preselection. Figure 4.11 shows all 224 channels with all the channels that have shown LCO activity in the spectrogram highlighted in red.

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Figure 4.11: All 224 bolometer chan- nels that are used. Those channels where LCO activity was seen in the spectrogram are highlighted in red.

However, the signal could be detected at any point of the sightline for each channel. To get a two dimensional structure it is essential to calculate the intersection points of the LOS. Since only the starting and endpoint of the lines of sight are known, linear equations are established to figure out the intersection points. Considering only the intersections of all highlighted LOS (red), figure 4.12 is received.

The next step is to compute the cross-correlation between those signals that are intersecting each other in order to quantify the common signal content for all intersecting LOS. Finally, the in- tersections and the corresponding maximum cor- relation value get matched and the latter is added color-coded to the plot in figure 4.13(a). Obvi- ously, many intersections have a quite low MCV (blue dots), meaning that the signals are not correlated. Hence, it is advisable to neglect all the intersections with an relatively low MCV. The higher the threshold is chosen the higher is the probability that both LOS were measuring the same signal. Taking an appropriate criti- cal lower threshold value of 0.5 leads to figure 4.13(b), where the localization can be recognized clearly. The vast majority of dots is found at the bottom of the vessel and a small region on the high field side (HFS) which is on the left in the plots of the vessel cross section.

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Figure 4.12: Intersections of all red highlighted lines of sight from figure 4.11.

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(a) Without threshold (b) MCV ≥ 0.5 (c) MCV≥0.5 and τ ≤20 µs

Figure 4.13: Intersection points with the maximum correlation value as a third dimension for discharge #31494: (a) without threshold for the MCV, (b) with a MCV-threshold of 0.5 and (c) with the same threshold and an additional restriction for the time shift (τ ≤ 20 µs). The points form a V-shaped area above the X-point.

In these cases, it is rather certain, that the same signal was measured, but it is still possible that the channels were measuring the signal at two different points on their sightline and not at the intersection point. Figure 4.14 illustrates this problem. Both channels were measuring LCO activity, therefore have a sufficiently high max- imum correlation value. The calculation of in- tersection points delivers the red dot, but the origin of the correlated signal could be at two different points along the lines of sight indicated by the black crosses. Such misleading intersec- tion points are still mapped in figure 4.13(b). However, to get the spatial structure it is es- sential to know where in the plasma the origin of the measured signal was. Consequently, it is necessary to restrict the area close to the inter- section of two LOS. Fortunately, the correlation analysis provides us both - the correlation value as well as the time displacement τ of the MCV that is corresponding to a phase shift between the signals. Obviously, if there is no time displacement the correlating activity were measured at the same time on both channels meaning at the intersection point. To be sure, that the signal origin was from the intersection point or at least a small area around, the time lag τ should be under a distinct limit. The smaller the threshold value is chosen the closer the signal origin is to the intersection. A reasonable value is τ = 20 µs because it corresponds to a maximum phase shift of approximately Δφ = 0.1π for a LCO-frequency of fLCO = 2.5 kHz[1]. Only plotting those intersec- tions where the maximum time shift is smaller than or equal to this limit leads to figure 4.13(c). These are all points where two bolometer channel were measuring the same signal or at least a signal with low relative phase, meaning the LCO activity was near the intersection point.

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Figure 4.14: Two channels with LCO activity could lead to a wrong intersection. Their maximum cor- relation value is high, however, it is possible that they have measured the LCO signal at two different points in the plasma.

Now, the localization is even more impressive. The density of points near the X- point is quite high. Other points build lines along the separatrix on the high as well as on the low field side (LFS) or at least very close to it. The accumulation of these points forms a V-shaped area above the X-point.

However, high correlation values do still not definitely mean that this was caused by a limit-cycle oscillation. It is possible, that both channels are correlated but due to another - maybe parasitic - frequency. To get rid of this constraint it is necessary to consider the coherence, providing the correlation as a function of frequency.

4.3.2 Coherence

4.3.2.1 Cross Power Spectrum and Phase

Calculating the CPS, coherence and phase as described in 3.2.2.2 results in figure 4.15 as an example. For reasons of clarity only the first 10 kHz are shown. In the given example the LCO-frequency is clearly visible in the Cross Power Spectrum (a) as well as in the coherence (b) as a dominant peak at approximately 1.6 kHz.

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Figure 4.15: (a) Cross Power Spectrum, (b) coherence and (c) phase between two channels of discharge #31494. The CPS as well as the coherence show the same peak at the LCO-frequency of approximately 1.6 kHz.

4.3.2.2 Spatial structure

Now, the corresponding frequency to the maximum coherence value is identified. Comparing this frequency with the one found in the spectrogram (cf. figure 4.5) tells if the coherence between both sig- nals indeed is at the LCO-frequency. Thus, the coherence for every remain- ing intersection point in figure 4.13(c) is calculated and the corresponding fre- quency to the coherence maximum is de- termined. Each intersection, where this frequency is not equal to the LCO- frequency is neglected in figure 4.16

In this case only two points disappeared in comparison with fig. 4.13(c) due to taking a really constant I-phase where the correlation is almost obvious or not. However, generally this step is necessary to be definitely sure that all the depicted points are areas where limit-cycle oscil- lations in the plasma were measured by the bolometer diodes. The final points are all close to the separatrix and the X-point, showing nearly the same V-shape. Espe- cially the red dots on the left-hand side indicate a movement along the separatrix. The reason for having no points on the top of the vessel is that for every intersection two channels with a signal are required. There are fewer intersections of LOS on the one hand, and, on the other hand the intensity is lower on the top (see section 4.2.2) due to lower neutral gas content and therefore lower line radiation. Moreover, the sensitivity of the bolometer diodes depends on the photon energy. For some reason it is possible that a temperature gradient within the plasma in vertical direction exists. Subsequently, the energy of the emitted photons vary and therefore the strength of the LCO signal. Unfortunately, the dependency of the diode sensitivity is particularly high in the low energy spectrum (cf. section 2.2.2.1).

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Figure 4.16: All points of discharge #31494 where LCOs certainly occur. It is clearly visible that the vast majority of points form a V-shaped region close to the X-point and the separatrix.

[...]

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Details

Title
Spatial Structure of Limit-Cycle Oscillations
College
Technical University of Munich  (Max-Planck-Institut für Plasmaphysik)
Grade
1,3
Author
Year
2016
Pages
129
Catalog Number
V342545
ISBN (eBook)
9783668330504
ISBN (Book)
9783668330511
File size
7848 KB
Language
English
Notes
The source code (written in Python) that was developed and used during this work is included in the appendix.
Tags
Plasmaphysik, Kernfusion, Fusion, ITER, ASDEX, ASDEX Upgrade, IPP, Max-Planck-Institut, Fusionsplasma, I-phase, Limit-cycle oscillation, Bolometer, Korrelation, Kohärenz, Räumliche Struktur, Lawson Kriterium, Deuterium, Tritium, Spektrogramm, Abtastfrequenz, Nyquist-Shannon Theorem, Plasmaheizung, DEMO, Fourier Analysis, L-mode, H-mode, Photodiode, Bachelorarbeit, TUM, Technische Universität München, Ulrich Stroth
Quote paper
Bernd Kohler (Author), 2016, Spatial Structure of Limit-Cycle Oscillations, Munich, GRIN Verlag, https://www.grin.com/document/342545

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