Frequency Scaling of Limit-Cycle Oscillations

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Frequency Scaling of Limit-Cycle Oscillations

Frequenzskalierung von Räuber-Beute-Modellen

A Bachelor Thesis in Physics by Alexander van Roessel

Technical University Munich Physics Department

15.07.2016

First Reviewer: Prof. Dr. U. Stroth Second Reviewer: Prof. Dr. R. Kienberger Supervisor: Dr. G. Birkenmeier

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Introduction

For centuries, there has been an increasing energy demand in the world. This trend will most likely proceed in the near future. At the same time it has become clear that the carbon dioxide emission, which arises from the consumption of fossil resources such as oil and coal, harms the global climate [1]. Therefore it is wide consensus that energy production using fossil resources must be replaced by other technologies, which do not emit greenhouse gases or harm the environment in another way. Since most of the renewable energies, such as wind- and solar energy, are very volatile and there is not yet a sucient storage technology in sight, it is important to nd a new energy source which can carry the base load of the power supply system. One promising technology which might be able to meet these requirements is nuclear fusion (cf. sec. 2.1).

Among the various fusion technologies, that are investigated all over the world, a very promising one is fusion using magnetic connement. There are two basic set-ups for this technology, the Tokamak, which will be introduced in sec. 2.2, and the Stellarator [2].

The data considered in this thesis was obtained at ASDEX Upgrade, a fusion experiment conducted by the Max-Planck-Institut für Plasmaphysik (IPP) in Garching, Germany, which is a Tokamak set-up.

In fusion plasmas, a desirable regime of high connement can occur if sucient heating power is injected into the plasma [3]. Since an improved connement in fusion plasmas can contribute to the realisation of a fusion reactor and enhance its possible eciency, it is worthwhile trying to understand the physics behind this phenomenon. At the transition into this regime of high connement, the so called I-phase, which is introduced in sec. 2.4, arises and shows characteristic oscillations of density, magnetic elds, turbulence levels and ow velocity in the plasma edge, which can be modelled by limit-cycle oscillations.

It is believed that the comprehension of these limit-cycle oscillations (LCOs) will help to understand the transition into higher connement regimes.

In this thesis, I will examine the frequency scaling of the LCOs in the I-phase and compare it to a tted scaling proposed by G. Birkenmeier et al. [4]. In order to do so, I will consider various plasma parameters and congurations and make use of two dierent methods for the determination of the frequency (cf. sec. 3.1 and 3.2). It will turn out that a slightly adapted scaling applies quite well for practically all of the investigated plasma types and congurations. The examined plasmas will encompass dierent isotopes such

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Chapter 1 Introduction

as Helium, Deuterium and Hydrogen, upper and lower single null conguration (cf. sec.

2.2), as well as transitions from high to low connement (H-L transitions) and vice versa.

Apart from the frequency scaling of the LCOs, the heating power dependence of the occurrence of the I-phase will also be investigated. In addition, several further observations will be discussed.

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Theoretical Background

2.1 Nuclear Fusion and Plasma

The main energy source of our solar system is the sun. The sun generates the energy by nuclear fusion. Thus, the sun makes use of the binding energy contained in the nuclei of the atoms, which is released when two nuclei merge. It produces a multiple of the energy per nucleon that can be extracted from nuclear ssion which is applied in conventional nuclear power plants.

In order to achieve nuclear fusion, the Coulomb repulsion between two nuclei must be overcome. In order to full this condition only by thermal movement of the atoms, extremely high densities and temperatures up to 150 million Kelvin are required. At those conditions, every substance is in the plasma state, which is one of the four fundamental states of matter. The plasma state is characterised by the fact that electrically charged particles are, at least partially, unbound which allows them to interact with electromag- netic elds although the substance as a whole appears neutral. For a comprehensive introduction to plasma physics in general and a more precise denition of the plasma state see [5].

There are various approaches to how to reach the extreme conditions that are needed to conduct fusion. One of them is to conne the substance supposed to perform nuclear fusion with magnetic elds. A detailed description of this technique can be found in [6].

One of the two prevalent set-ups for fusion by magnetic connement is the Tokamak, the other one is the Stellarator [7]. The Tokamak will be outlined in sec. 2.2. At ASDEX Upgrade, which is a Tokamak, Hydrogen, Deuterium and Helium plasmas are investigated.

Most of the international fusion research focuses on the isotopes of Hydrogen and Helium because the fusion of light elements yields the largest amount of energy per mass and is thereby favourable for the commercial use of fusion energy.

2.2 The Tokamak

As mentioned before, the Tokamak is one of the two convenient set-ups for a fusion reactor featuring magnetic connement. The principle of a Tokamak is displayed in g. 2.1. The

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Chapter 2 Theoretical Background

toroidal shape is characteristic for this set-up. The magnetic connement of the plasma is achieved by the combination of a poloidal and toroidal magnetic eld which result in helical magnetic eld lines. In a Tokamak, toroidal coordinates indicate the position in the torus and poloidal coordinates indicate the position in the cross section for a given toroidal position. The ratio between the number of revolutions of the magnetic eld lines in toroidal direction and in poloidal direction is called the safety factor and depends on the radius. Typically, one considers the safety factor at 95% of the minor radius, so q₉₅, in order to compare dierent plasma congurations. The term 'safety' is used due to the resulting stability of the plasma for high values ofq95.

The poloidal magnetic eld is produced by a current inside the plasma itself, which again is induced by a transformer circuit in the centre of the torus. In order to induce the plasma current, the magnetic eld of the transformer circuit and therefore the current inside this circuit need to continuously increase. Thus, the maximum transformer current determines the time the connement can be maintained. This is why Tokamaks can only be operated in pulses.

The toroidal magnetic eld is produced by a number of eld coils that run concentrically around the torus. In addition, two large poloidal eld coils alongside the torus are employed to adjust the position and the shape of the plasma to the experimental requirements.

The helical magnetic eld lines do not close after one revolution but form so called ux surfaces. These ux surfaces are indicated by the dashed lines in g. 2.2, which depicts the prole view of the torus. They are also used to dene a relative radial coordinate, the normalised poloidal ux coordinateρ_pol, which is constant along a ux surface. The ux surface withρpol = 1is called the separatrix and is the last closed ux surface. Magnetic eld lines atρ_pol >1do not form magnetic ux surfaces anymore and end on the wall or in the so called divertor, which is an area at the bottom of the torus that has a special design to withstand the increased exposure to heat and plasma particles.

In the normal lower single null (LSN) conguration, which is depicted in g. 2.2, the X-point, which is the intersection of the separatrix with itself (cf. g. 2.2), lies close to the lower divertor. There is another divertor at the top of the torus which is used when the fusion plasma is operated in upper-single-null (USN) conguration. That is, when the plasma shape is inverted so that the X-point lies at the top, close to the upper divertor. Both congurations are shown in g. 2.3.

In both LSN and USN conguration it can be distinguished between favourable and unfavourable conguration. Favourable conguration means, that the ion grad-B drift, which is dened by

vi,∇B= W⊥

q

B× ∇B

B³ (2.1)

where B is the magnetic eld, W⊥ the kinetic energy of the ions perpendicular to the

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Figure 2.1: Illustration of the Tokamak set-up. The actual plasma has the shape of a torus and is coloured in pink. The primary transformer circuit in the centre induces the plasma current which then produces the poloidal magnetic eld (green). The toroidal eld coils, which run concentrically around the plasma, produce the toroidal magnetic eld (blue). Together, the poloidal and toroidal magnetic eld result in a helically shaped magnetic eld (black lines). There are also some additional poloidal eld coils (grey) which are used to adjust the plasma shape and position. Picture taken from [8].

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Chapter 2 Theoretical Background Figure 2.2: Prole view of

the ASDEX Upgrade torus.

All in-vessel components are coloured in grey. The ux surfaces are indicated by the black dashed lines and the separatrix, the last closed ux surface, is coloured in red. One can also see, that the ux surfaces with ρ_pol >

1, that are not closed, end on the lower divertor. The intersection of the separatrix with itself is called X-point.

1.00 1.25 1.50 1.75 2.00 2.25 2.50

-1.0 -0.5 0.0 0.5 1.0

Upper Divertor

Lower Divertor Flux Surface Separatrix

X-Point

magnetic eld and q the charge of the particle, points towards the X-point. In unfavourable conguration, it points in the opposite direction. Thus, to stay in favourable conguration when changing the plasma shape from LSN to USN one has to invert the magnetic eld B at the same time.

2.3 Diagnostics

2.3.1 Mirnov Coils

Apart from the plasma current, which is externally induced, other currents in the fusion plasma naturally occur giving rise to time-dependent magnetic elds. The variation in the magnetic eld B˙ can be determined by so called Mirnov coils. If the magnetic eld along the Mirnov coils changes, it induces a voltage in the coils which can be measured.

With an appropriate calibration,B˙ can be calculated directly from the raw signal given in Volts.

At ASDEX Upgrade, there are 32 Mirnov coils applied at poloidal positions around the cross section of the vessel [9]. Due to reasons explained in sec. 2.4, I will use theB˙ signal for the determination of the frequency of the limit-cylce oscillations. Since the visibility of these oscillations highly depends on the poloidal position of the respective Mirnov coil [4], I decided to use the coil that is located right below the lower divertor (technical

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1.00 1.25 1.50 1.75 2.00 2.25 2.50 -1.0

-0.5 0.0 0.5 1.0

#33724 5.1 s

(a) LSN conguration.

1.00 1.25 1.50 1.75 2.00 2.25 2.50

-1.0 -0.5 0.0 0.5 1.0

#30866 4 s

(b) USN conguration.

Figure 2.3: Cross section of the Tokamak in lower single null conguration (a) and upper single null conguration (b). The red line indicates the separatrix atρpol = 1.

label: MHE/C09-23) because the signal was strongest there. For discharges in USN conguration, I employed the respective coil above the upper divertor (MHD/C09-09).

The positions of both coils are shown in g. 2.4.

2.3.2 Thomson Scattering

In the core plasma, all Hydrogen atoms are ionised so that only free electrons and no bound electrons have to be considered when examining the scattering of electro-magnetic radiation in the plasma. Since the ions are a few thousand times heavier than the electrons, light is mainly scattered by the electrons. The non-relativistic scattering of light by free electrons is called Thomson scattering.

When a moving charged particle scatters monochromatic light, the Doppler eect leads to a spectral shift of the scattered radiation. Since it depends on the velocity of the particle relative to the detector, it is possible to reconstruct the velocity distribution of

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Chapter 2 Theoretical Background Figure 2.4: Cross section of

the ASDEX Upgrade torus.

The red line illustrates the separatrix and the two green straights represent the lines of sight of the interferometer chords. The H5 interferometer chord measures the line averaged edge density and the H1 interferometer chord the line averaged core density. The line of sight of the Thomson scattering is indicated by the blue dashed line. The two purple dots represent the positions of the Mirnov coils.

1.00 1.25 1.50 1.75 2.00 2.25 2.50

-1.0 -0.5 0.0 0.5 1.0

Mirnov Coil (MHD C09-09)

Thomson Scattering

Mirnov Coil (MHE C09-23)

DCN Interferometer (H5)

DCN Interferometer (H1)

the electrons from the spectrum of the scattered light. In the set-up at ASDEX Upgrade, it is possible to measure the electron temperature Te and the electron density ne with the Thomson scattering system, since it is related to the velocity and the number of the electrons. The line of sight of the Thomson scattering device is indicated by the blue dashed line in g. 2.4. The laser is red from the bottom and the light is detected at several dierent vertical positions in order to get proles ofTe and ne. A more detailed description of the Thomson scattering system at ASDEX Upgrade can be found in [10]

and [11].

2.3.3 DCN Interferometer

The principle of the density measurement by interferometry is that a plasma has a re- fractive index smaller than 1, which means that the wavelength is greater inside the plasma than in vacuum. This causes a phase shift of the light when it travels trough the plasma. The total phase shift is proportional to the line-integrated electron density along the whole path of the light. Thus, if the total phase shift of a laser beam is measured via interferometry, one can determine the line-averaged electron density.

At ASDEX Upgrade, a Deuterium-Cyanide (DCN) laser with 5 dierent channels is used. The dierent channels (H1 to H5) feature dierent lines of sight in the plasma. In this thesis, I will regard the H1 channel, which delivers the density in the plasma centre, and the H5 channel, which delivers the density at the plasma edge. Both of them are

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displayed as green lines in g. 2.4. The H5 interferometer chord is used to measure the line averaged electron edge densityn¯e,edge and the H1 interferometer chord is applied to determine the line averaged electron core densityn¯n,centre.

2.4 The I-phase

As mentioned in the introduction, a regime of high connement arises in fusion plasmas if sucient heating power is injected into the plasma [3]. At the transition from the regime of low connement (L-mode) to the regime of high connement (H-mode) and vice versa, the I-phase occurs.

In general, the L-mode can be characterised by low connement of energy and particles.

It is accompanied by a high level of turbulence in the plasma and it only sustains small gradients of for example density and temperature. On the contrary, the H-mode features a low turbulence level and high energy connement. Furthermore, in H-mode larger gradients can be achieved, especially in the region close to the separatrix, where a sort of transport barrier for both particles and energy is formed. This is very convenient and might help to realise a self-sustaining fusion process.

During the I-phase, hence between the L-mode and H-mode, characteristic oscillations, the limit-cycle oscillations, occur which can be described by a set of coupled linear dierential equations (cf. sec. 2.5). The LCOs occur close to the separatrix, thus in the same region as the transport barrier, which is already established during the I-phase. This suggests that they are related to the desirable improved connement. The examination of the frequency scaling of these oscillations based on experimental data is subject to this thesis.

The LCOs can be found in many dierent signals, such as: the Doppler reectometer frequency shift f_D, the Doppler reectometer amplitude A_D, the divertor current I_div, the variation in the poloidal magnetic eld B˙_pol, referred to as B˙ in the following, and the electron densityneatρ_pol = 0.998. These signals are illustrated in g. 2.5 for a short time range in a typical I-phase. The signal I used for the determination of the frequency is the variation in the magnetic eld B˙ because the diagnostics for this signal are very reliable and the LCOs occur very regularly and neatly, as can bee seen in the illustration (g. 2.5).

2.5 Limit-Cycle Oscillations

In Biology, predator-prey models were introduced as the so called Lotka-Volterra equations and are used to describe the development of both a predator and a prey population which depend on each other. These models are also applicable for the description of the oscillations in the I-phase [12] [13]. In this case, the shear ow intensity can be identied with the predators and the turbulence energy in the plasma with the prey. The

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Chapter 2 Theoretical Background

Figure 2.5: Illustration of the LCOs during the I-phase in various plasma signals: the Doppler reectometer frequency shiftf_D (a), the Doppler reectometer amplitudeA_D (b), the divertor current Idiv (c), the variation in the poloidal magnetic eld B˙pol (d) and the electron densityn_e at ρ_pol = 0.998(e). Image section taken from [4].

latter would actually continuously increase, but the shear ow intensity diminishes the turbulence energy [14]. Since the shear ow intensity again is driven by the turbulence itself, this situation is modelled with a set of coupled linear dierential equations of the Lotka-Volterra type.

Predator-prey models basically have two dierent types of solutions. On the one hand, there are single step transitions for which the predator eect dominates and all variation stops after an initial phase of growth. This would result in unclosed paths in the phase space with a x point. On the other hand, also stable cyclical solutions are possible for which the interaction of the two quantities causes an oscillating state of coexist- ence. These oscillations relate to closed paths in phase space and are called limit-cycle oscillations.

An extensive discussion about the theory of limit-cycles in general can be found in

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[15]. Limit-cycles in predator-prey models are covered by [16]. The experimental results of LCOs at the connement transition in fusion plasmas are summarised in [17].

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Chapter 3 Methodology

Both the∆tand spectrogram method for the determination of the I-phase frequency have been implemented in the programming language "Python" [18] and executed on ASDEX Upgrade servers. In order to determine the frequency of an I-phase, I rst isolated the relevant time period, in which the I-phase occurs in the magnetic signal (cf. sec. 2).

Subsequently, I applied the methods onto that tailored raw signal.

3.1 ∆t Method

The basic idea of the∆tmethod is that the local frequency can be calculated by f = 1

∆t, (3.1)

where∆tis the time between two I-phase bursts which appear as peaks in the magnetic signal.

In order to identify the peaks, I have determined a threshold ofS = 1 + 0.8·σ whereσ is the standard deviation of the magnetic signal during the I-phase. For a few exceptional discharges, the threshold had to be adapted because the amplitude of the signal was too high. In all other cases, the choice of the threshold S has proven itself very robust.

If the signal is above the threshold, it is considered as a peak and the starting and ending point of each peak is determined. The time between two starting points of ad- jacent peaks is then viewed as ∆t. Thus, the local frequency between two peaks can be calculated according to eq. (3.1).

A typical result of the frequency determination via the ∆t method is displayed in g. 3.1. The black line indicates the threshold S partly calculated with the standard deviation of the blue signal, which represents the variation of the magnetic eld. The green and red dots are the starting and ending points of the respective peaks. The rst and the last peak is omitted by the procedure, which is due to the implementation in the python code. Since the information loss is limited, this is an acceptable constraint.

In g. 3.1b, it can be seen that the signal is below the threshold most of the time.

This is characteristic for the LCOs in the magnetic signal. It turned out that for most

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3.43 3.44 3.45 3.46 t (s)

−2 0 2 4 6 8

_B (T/s)

#29303

(a)∆tmethod for a whole I-phase of discharge

#29303.

3.4435 3.4440 3.4445 3.4450

t (s)

−2

−1 0 1 2 3 4 5 6

_B (T/s)

#29303

(b) ∆t method for 3 peaks.

Figure 3.1: Outcome of the ∆t method for discharge #29303. The blue signal is the variation of the magnetic eldB˙ and the black line represents the threshold S which denes what is detected as a peak. The green and red dots indicate the starting and ending points of a single peak.

of the investigated I-phases (43 out of 45), the relative amount of time, during which the signal is above the threshold, lies between 10% and 25% on average. The remaining two I-phases just deviated a little from this range.

3.2 Spectrogram

Since the spectrogram is one of the many applications of the Fourier transform, I will initially give a short introduction to the latter and briey explain the fast Fourier transform.

Subsequently, I will outline the spectrogram method for the frequency determination.

3.2.1 Fourier Transform

The Fourier transform is basically a transformation from the time domain to the frequency domain. The idea is to nd the contributions of sinusoidal functions with dierent frequencies to an arbitrary periodic signal. It can be calculated with the following formula:

F(ω) =

∞

Z

−∞

f(t)e^−iωtdt . (3.2)

In mathematics, it can be shown that generally every signalf(t)that is depicted in the

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Chapter 3 Methodology

Figure 3.2: Illustration of the Fourier transform. At the top left, the signal is shown in time domain. On the right, the sines at various frequencies, of which the original signal consists, are displayed. They make up the spectrum of the frequency, thus the representation of the signal in frequency domain, which is depicted at the bottom left.

Picture taken from [21].

time domain can also be depicted in the frequency domain byF(ω). It is also possible to prove that both representations are equivalent and contain the same information.

In gure 3.2, the procedure of the Fourier transform is illustrated. The signal, initially in time representation, is modelled with sines and cosines at various frequencies. All of these frequencies contribute dierently to the signal according to their Fourier coecients, which make up the so called spectrum of the signal. This spectrum can be derived with the aid of eq. (3.2). For a vivid introduction to the Fourier transform consider [19] or [20].

A specic algorithmic implementation of the Fourier transform is the fast Fourier transform (FFT). It reduces the complexity of the computation of the transform fromO(n²)to

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O(nlogn) and thereby distinctly decreases its computation time [22]. A comprehensive description of the FFT can be found in [23].

3.2.2 Spectrogram Method

The spectrogram method makes use of the Fourier transform that has been introduced in the previous section. It divides the magnetic signal during the I-phase into a number of overlapping subsections (steps) and calculates the spectrum of the raw signal for each subsection. One has to predene the number of data points between two consecutive evaluations of spectra (nstep), which determines the step width. It also denes the number of steps that are made in total, depending on the total length of the considered time trace. In the case at hand, the step width was set to 64 data points.

The number of samples considered per step (nt) for the determination of the spectrum also had to be chosen manually. Since it denes the frequency resolution, it is ought to be as large as necessary but at the same time as small as possible because it directly aects the runtime of the program. It turned out that an nt of 16384 applied best for my analysis. If one also considers the resolution of the raw signal B˙, one can derive that the frequency resolution of the spectrogram is approximately 121.3 Hz. This should be considered when interpreting results obtained with the spectrogram method. Both spectrogram parameters were chosen in powers of 2 because this way the algorithm works fastest.

With this algorithm, one can now calculate the spectrum for every step and thus obtain the spectrogram. Fig. 3.3 shows the spectrogram of the magnetic signal of discharge #29306. The frequency is plotted against the time, respectively the steps, and the colour as a third dimension represents the particular spectrum. The brighter the colour, the larger is the contribution of that frequency to the spectrum.

As a next step, I determined the maximum of the spectrum for every step, indicated by the black line in g. 3.3, under a number of conditions which are implemented to ensure the correct determination of the frequency. This results in the main frequency of the signal, i.e. the dominant frequency in the spectrum. In g. 3.3, one can also recognise the higher harmonics at an integer multiple of the main frequency, particularly at the beginning of the I-phase, which have to be excluded by means of the frequency determination in order to nd the fundamental frequency.

As the black line agrees very well with the fundamental frequency, it can be stated that the spectrogram method works very well with regards to the frequency determination.

In practice, it has overall proven itself very reliable.

3.3 Comparison of Methods

Both the ∆t and spectrogram method have some advantages and disadvantages. The

∆t method for instance yields a local frequency for every peak and does not average

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Figure 3.3: Spectrogram of the magnetic signal for discharge #29306. One can identify the major frequency starting at 2 kHz and decreasing to about 1 kHz as well as some higher harmonics. The black line indicates the maximum of the spectrum found by the algorithm for each step. As one can see, it coincides very well with the actual base frequency I want to derive.

over a longer period of time. That way, one does not lose any information due to the averaging process in contrast to the spectrogram. This results in a higher resolution of the frequency, which allows to resolve smaller variations and larger gradients of the frequency. In addition, with the∆tmethod one can not only derive the frequency of the I-phase but also determine the height of the peaks and the time the signal is above the threshold. I will make use of this additional information in sec. 4.7.1. On the other hand, the peak detection is relatively unreliable because I determined the threshold once and, with a few exceptions, applied it to all I-phases. As a result, the threshold is too high for some peaks, especially in unsteady I-phases, so that the frequency is not detected correctly. This leads to unsolicited scattering in the plot.

The main advantage of the spectrogram method is its high reliability, which is due to the averaging process (cf. sec. 3.2.2). Small irregularities do not aect the outcome of

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3.45 3.50 3.55 3.60

t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

f ( k H z)

#29306

^{¢ t}spectrogram

Figure 3.4: Comparison of the spectrogram and ∆t method for discharge #29306. It is obvious that both practically yield the same result in relation to the frequency determination. It is characteristic for the I-phase to have a nice decrease in the frequency at the beginning. In this case, it starts at 2 kHz and drops down to 1 kHz before it continues at nearly constant frequency.

the spectrogram in contrast to that of the ∆tmethod. Additionally, the averaging also decreases the general scattering in the plots. However, the spectrogram can only identify a discrete number of frequencies with a resolution of 121.3 Hz (cf. sec. 3.2.2) as against the∆tmethod which can detect arbitrary frequencies.

Fig. 3.4 shows, that, at least for nice I-phases, both methods coincide very well and yield the same result in relation to the frequency. This plot also features a very common behaviour of the I-phase: the frequency decreases at the beginning and then proceeds nearly constant. For I-phases at transitions from high to low connement regimes, this process happens reversely.

Summarising, I will employ the spectrogram method for most of my investigations, due to its superior reliability and reduced scattering. The only exceptions will be examina- tions, where it does not provide the necessary information, in which case I will apply the

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∆tmethod. For example, the frequency-amplitude correlation analysis in sec. 4.7.1 can only be conducted with the ∆t method.

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Frequency Scaling of the I-Phase

In this chapter, I will refer to the frequency scaling of the I-phase proposed by G. Birken- meier et al. in [4]. I will conrm their ndings with both the ∆t and the spectrogram method (cf. sec. 3.1 and 3.2) and show that they yield the same result. Subsequently, I will examine the frequency scaling in various types of plasmas, including Helium and Hydrogen plasmas, USN-conguration as well has H-L-transitions, and I will show that the scaling applies for them as well.

For reasons of clarity, I have limited the peaks that are used for the frequency scaling analysis to the rst 60 and for H-L transitions to the last 60 respectively. If more peaks are considered, the plots just get more confusing and it would be harder to extract the relevant information. The results essentially stay the same since most of the variation happens at the beginning of the I-phase anyway (cf. sec. 3.3).

A number of examined discharges had to be discarded due to several reasons. In some discharges, the magnetic X-point location was too close to the divertor tiles compared to standard congurations. In others, the plasma current was still varying so that one can not assume the plasma to be in an equilibrium. Additionally, some diagnostics, for example the Thomson scattering system for the electron temperature, were not available for a few discharges or exhibited non-physical behaviour. In contrast, the applied methods for the frequency determination have proven themselves very robust.

Table 4.1 is a comprehensive collocation of the discharges examined throughout this thesis along with their plasma type, conguration and other important plasma parameters.

4.1 Conrmation of Previous Findings

In [4], G. Birkenmeier et al. have conducted a multivariate linear regression for the dependence of the LCO frequency on various variables. The outcome of this analysis was the following formula:

f_{f it} = 976,031 ¯n^−1.10_e,edge T_e^−0.91 B_t^1.93 q^−1.36₉₅ (4.1) where f is the predicted frequency in Hz, n¯ [10¹⁹m⁻³] the line averaged edge

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Chapter 4 Frequency Scaling of the I-Phase

discharge isotope conguration transition B_t (T) I_p (MA) q₉₅

#29302 D LSN L-H -2.5 0.60 6.85

#29303 D LSN L-H -1.8 0.60 4.96

#29306 D LSN L-H -1.4 0.60 3.85

#29307 D LSN L-H -1.4 0.60 3.85

#29308 D LSN L-H -3.2 0.60 8.72

#29309 D LSN L-H -2.5 0.83 4.90

#29310 D LSN L-H -1.4 0.47 5.00

#29311 D LSN L-H -1.8 0.60 4.94

#29312 D LSN L-H -1.8 0.60 4.96

#29315 D LSN L-H -3.2 1.07 4.96

#30866 D USN L-H 2.5 1.00 3.90

#32927 D USN L-H 2.5 1.00 4.10

#32924 D USN L-H 2.5 1.00 3.90

#30863 D unfavourable USN L-H -2.5 1.00 3.90

#27124 D LSN H-L -2.4 1.00 4.00

#27126 D LSN H-L -2.4 1.00 4.00

#27129 D LSN H-L -2.4 1.00 4.00

#29120 H LSN L-H -2.3 0.80 4.70

#29125 H LSN L-H -2.3 1.00 3.80

#29126 H LSN L-H -2.3 0.80 4.70

#31397 H unfavourable USN L-H -2.5 0.80 4.90

#31399 H unfavourable USN L-H -2.5 0.80 4.90

#32719 He LSN L-H -2.5 0.80 5.00

#32723 He LSN L-H -2.5 0.80 5.20

#32785 He LSN L-H -2.5 0.60 6.70

Table 4.1: Collocation of all discharges considered in this thesis. Included plasma parameters are the toroidal magnetic eld B_t, the plasma currentI_p and the value of the safety factorq₉₅.

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electron density,T_e [eV] the electron temperature, B_t[T] the toroidal magnetic eld and q95 the safety factor which has been introduced in sec. 2.2.

I have made two adaptations to this formula. On the one hand, I had to replacen¯_e,edge by n¯_e,centre [10¹⁹m⁻³] due to reasons explained below. On the other hand, I introduced a correction factor c in order to adjust the scaling to my ndings and also compensate for the deviation which occurs because I employed the core density instead of the edge density as proposed by G. Birkenmeier et al. The resulting formula for the scaling is:

f_scal =c· 976,031 ¯n^−1.10_e,centre T_e^−0.91 B_t^1.93 q₉₅^−1.36 . (4.2) In order to calculate fscal according to eq. (4.2), I had to acquire the values of the dierent plasma quantities, which are given as data time traces stored in the so called shotle system of ASDEX Upgrade. In case of the electron temperature, which is given as a radial prole, I decided to acquire it atρpol= 0.95 by linear interpolation since the I-phase occurs at the edge of the plasma, close to the separatrix atρ_pol = 1. I tried to obtain all values at the same time as the beginning of each spectrogram step (cf. sec.

3.2.2) or the beginning of each peak in case of the ∆t method. If the value was not available at that exact time, I interpolated linearly in order to obtain the value at the desired time point. Therewith, I was able to determine the theoretical value given by eq.

(4.2) and compare it with the measured frequency from the experimental data.

I have plotted fscal against the frequency for the same 10 discharges as in [4] which are all Deuterium plasmas in normal conguration. They include I-phases at various dierent plasma currents and magnetic elds (cf. tab. 4.1). If the scaling of eq. (4.2) is correct, one would expect that the data points all lie on the bisectrix. The result is displayed in g. 4.1 where one can see that the points make up a neat line, which lies slightly above the bisectrix on average but has the correct slope of one. This result approves the considered scaling. From now on, I will employ the correction factor as c = 1.58. By this choice, the experimental data agrees best with the predictions made by the scaling in eq. (4.2). In the following, the ten discharges introduced in this section will act as reference for my further analysis of the eects of dierent plasma types and congurations on the frequency scaling.

In eq. (4.2) it actually would make sense to employ the line averaged edge density

¯

ne,edge, measured with the H5 interferometer, because the I-phase emerges at the edge of the plasma. But since I will also examine discharges in USN conguration, in which the line of sight of the interferometer would include the X-point region which would deteriorate the measurement, another interferometer chord should be used. Since I want to compare discharges in USN and normal conguration, I have to use an interferometer chord which is applicable in both congurations. This is why I decided to use the H1 interferometer chord, which measuresn¯e,centre.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f (kHz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz) _#29302

#29303

#29306

#29307

#29308

#29309

#29310

#29311

#29312

#29315

Figure 4.1: The frequency scaling of the I-phase for a set of 10 discharges. fscal has been calculated according to eq. (4.2). The fact that all points lie on one straight line, almost on the bisectrix which is illustrated by the grey line, strongly indicates that the proportionality, given by eq. (4.2), is applicable. The set of 10 discharges includes Deuterium discharges at various plasma currents and magnetic elds (cf. tab. 4.1).

The average factor, by which n¯_e,centre (H1-chord) deviates fromn¯_e,edge (H5-chord) is 1.319. Since the density occurs in the denominator of eq. (4.2), f_scal decreases by a factor of 0.737 on average. Apart from that, both interferometer chords approximately give rise to the same trend for f_scal as is shown in g. 4.2. Therefore it is legitimate to use the H1 instead of the H5 interferometer chord.

In g. 4.2, fscal,uncorrected, which has not been corrected by the dierence in density, was used in order to illustrate the dierence between the two chords. The densityn¯_e,centre measured by the H1 chord has to be halved in order to coincide with g. 4.1. Interestingly, the correction factor c would be one, thus not needed, if one assumes that the density

¯

n_e,centre is exactly twice as large as n¯_e,edge. This is what seems to be true at least for higher frequencies (cf. g. 4.2). However, as this does not apply for all frequencies, I will keep the correction factor asc= 1.58.

For the sake of completeness, g. 4.3 depicts the frequency scaling of the reference discharges for both the spectrogram and ∆t method. As one can see, the scaling does

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (kHz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal;uncorrected (kHz)

H5-Chord H1-Chord

Figure 4.2: Illustration of the dierence if fscal is calculated with the H5 (blue) or the H1 (red) interferometer chord (cf. sec. 2.3.3). In this case, fscal,uncorrected, which has not been corrected by the dierence in density, was applied. Apart from a factor of 0.737, both diagnostics result in a coherent frequency scaling and therefore it is appropriate to employ the H1 instead of the H5 interferometer chord for the density determination.

not depend on the method that has been applied, except for the increased scattering of the∆tmethod due to reasons outlined in sec. 3.3.

From now on, I will solely make use of the spectrogram method for the frequency scaling analysis because it scatters less, which makes the plots more clear and lets one extract the important information more easily.

4.2 Upper Single Null Conguration

Since there is no major dierence between the upper and lower single null conguration (cf. sec. 2.2) apart from the inverted shape, one would naively expect that it does not have any impact on the frequency scaling of the I-phase. As can be seen in g. 4.4, this is the case: the data points of the discharges in USN conguration agree very well with

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f (kHz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

Spectrogram

¢t - Method

Figure 4.3: Comparison of the frequency scaling of the I-phase depending on the method applied. f_scalhas been calculated with both the∆tmethod (blue) and the spectrogram method (red). One can see that both methods basically yield the same result except for the increased scattering of the∆tmethod, which is discussed in sec. 3.3.

those of the reference discharges. The index added to the discharge number after the underscore indicates the number of the I-phase in a discharge, in which several I-phases occurred. I will always apply this notation from now on.

It has to be emphasised that I-phases typically occur only in favourable conguration (cf. sec. 2.2). This is a very interesting observation since the power threshold to ac- cess the H-mode is lower in favourable conguration compared to the unfavourable one.

However, there are a few exceptional discharges in which the plasma enters the I-phase although the Tokamak is operated in unfavourable conguration. The blue data points at the bottom left in g. 4.4 originate from discharge #30863 which is the only one that had an unfavourable USN conguration. As one can see, even this discharge ts into the frequency scaling perfectly. I.e. although the L-H power threshold is higher and the probability that the I-phase appears is lowered in unfavourable conguration, the frequency scaling of the I-phase is the same as usual.

Overall, it can be stated that the intuitive assumption, that the frequency scaling of the I-phase should not exhibit an up-down asymmetry, has been conrmed.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (kHz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

Reference

Unfavourable USN

#30866_1

#32927_1

#32927_2

#32924_1

Figure 4.4: Frequency scaling of I-phases in upper single null conguration compared to the reference discharges in normal conguration. The points for all discharges in USN conguration coincide very well with the reference and lie on the bisectrix. The USN discharge in unfavourable conguration (#30863), which is illustrated by the blue diamonds, also ts into the scaling perfectly. The index added to the discharge number after the underscore indicates the number of the I-phase in a discharge, in which several I-phases occurred.

4.3 HL-Transition

So far, I-phases at the transition from L-mode to H-mode were examined. The frequency scaling appears to be applicable for both upper and lower single null conguration. Now we will investigate I-phases at the back transition from H- to L-mode.

The frequency scaling of the H-L transitions is depicted in g. 4.5. Some of the H-L discharges coincide with the reference discharges well, but others, mainly at lower frequencies, deviate a little from the scaling according tof_scal (eq. (4.2)). However, the deviation is not large.

Overall, g. 4.5 indicates, that the frequency scaling of the LCOs of H-L transitions deviate a little from those of L-H transitions. Thus, the physics of LCOs at the H-L transition might be somewhat dierent from that of an L-H transition. Other eects

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f (kHz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

Reference

#27124_1

#27124_2

#27124_3

#27126_1

#27126_4

#27129_3

#27129_4

Figure 4.5: Comparison of the frequency scaling of H-L transitions and the 10 L-H reference discharges (red). Some of the H-L discharges lie neatly in the range suggested by the reference discharges but others, especially at lower frequencies, deviate a little.

may be an issue and further quantities may have to be taken into account in order to appropriately describe the frequency scaling of LCOs at the H-L transition.

4.4 Isotope Dependence

Since the frequency scaling of eq. (4.2) seems to apply for Deuterium plasmas in LSN and USN, favourable and unfavourable conguration and even for I-phases at H-L transitions, it is of interest to know whether this holds true for other elements and isotopes. In this case, Helium (He) and Hydrogen (H) were considered.

As shown in g. 4.6, the scaling of the frequency for Helium and Hydrogen plasmas seems to dier from that for Deuterium plasmas. Except for one outlier, the I-phases for both Helium and Hydrogen lie on one line but distinct lower than those in Deu- terium plasmas. Thus, one can conclude that f_scal is not applicable for other elements or isotopes.

It appears that f_scal is roughly half of what it should be according to eq. (4.2). This

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (kHz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

Reference HeH

Figure 4.6: Frequency scaling of the I-phase for Helium and Hydrogen compared to Deuterium. One can see that the scalings for Helium and Hydrogen, apart from one outlier at the top, agree quite well but clearly dier from that for Deuterium. It seems that the value offscal lies at the half of what would be expected if the scaling for the reference discharges was assumed. This suggests thatf_scal must be revised in order to be applicable for other isotopes.

indicates that the scaling has to be revised. A factor of two has to be added for the scaling of Hydrogen and Helium while the scaling for Deuterium should stay the same.

This can be achieved by adding a factor of q²/m to the scaling, where q is the charge, given in units of the elementary charge [e], and m the mass, given in atomic mass units [u], of the nucleus of the respective element or isotope. The adapted frequency scaling would then be:

f_scal^∗ =c· 976,031 ¯n^−1.10_e,centre T_e^−0.91 B_t^1.93 q₉₅^−1.36 ·q²

m . (4.3)

With this revised f_scal^∗ , the data coincides with the scaling for Deuterium very well and lie neatly on the bisectrix as is displayed in g. 4.7. In this plot, an outlier at the top has been cut o due to conformity and clarity of the illustration. Apart from this, it

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f (kHz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f¤ scal (kHz)

Reference HeH

Figure 4.7: Frequency scaling of the I-phase for Helium and Hydrogen compared to Deuterium with an adaptedf_scal^∗ (eq. (4.3)). Apart from the outlier at the top, which has been cut o in this plot due to conformity, the revised scaling f_scal^∗ seems to be applicable for dierent isotopes an elements.

is indicated that the adapted formula for the frequency scaling of the I-phase is also applicable for Helium and Hydrogen plasmas.

In Helium there are 2 electrons related to every Helium ion. Thus, the ion density is half of the electron density. Thereby, the necessary factor of 2 in the adapted scaling for Helium could also be realised by employing the ion density instead of the electron density.

Nevertheless, the factor q²/m is more convenient since it also applies for Hydrogen.

However, other possibilities for the realisation of the factor can not be precluded.

The isotope dependence is a quite strong eect and may be an important aspect for a theoretical description of the LCOs.

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4.5 Unfavourable Upper Single Null Conguration in Hydrogen Plasmas

So far, it was shown that the frequency scalingf_scal^∗ according to eq. (4.3) is applicable for dierent plasma types as well as USN congurations in Deuterium plasmas. In this section, the behaviour of the frequency scaling will be investigated based on six I-phases from two discharges where both the plasma type and the conguration were modied simultaneously.

The result of this examination is displayed in g. 4.8. Despite the scattering in the plot, it is apparent that all of the data points lie in the same region but not on the bisectrix as one would expect following the scaling (eq. (4.3)). Though, if the original scaling f_scal according to eq. (4.2) without the additional factor of q²/m were applied, the data points would coincide with the trend of the other discharges. However, this would be in conict with the adaptations made to the scaling in sec. 4.4. Since the I-phases in the discharges in unfavourable USN conguration of Hydrogen plasmas were extraordinary scenarios anyway, this result is not clear. On the other hand, the deviation of the considered I-phases from the main trend might t to the indication of a kink in the frequency dependence, which is discussed in sec.4.7.2. Thus, further investigations should be conducted in order to clarify the behaviour of the frequency scaling of the LCOs in I-phases in this specic type of discharge.

In conclusion, the considered data indicates that the regarded scaling f_scal^∗ does not apply in this very specic type of discharge. The scaling seems to be not universally applicable and further adaptations might be necessary.

4.6 Summary

The outcome of the study of the frequency scaling is summarised in g. 4.8, where the anticipated frequency is plotted against the measured frequency. One can clearly identify a straight line which agrees with the bisectrix very well and that includes most of the considered plasma types and congurations. Merely some I-phases at H-L transitions deviate a little from the overall trend as has been discussed in sec. 4.3. Beyond that, it is apparent that the Hydrogen plasmas in unfavourable USN conguration do not t into the regarded scaling (cf. sec. 4.5).

However, it is remarkable that discharges in the range of #27000 to #34000, which have been conducted in a time span of several years, agree in the frequency scaling of the LCOs. It is indicated that the scaling f_scal^∗ according to eq. (4.3) is quite robust even in very dierent plasma types and congurations but does not hold true for multiple simultaneous modications of the plasma type and conguration, for example Hydrogen plasmas in USN conguration.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f (kHz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f¤ scal (kHz)

Reference USNHe

HH-L Transition Unfav. USN in D Unfav. USN in H

Figure 4.8: This gure shows the outcome of the frequency scaling analysis. Not all of the data points lie exactly on the bisectrix but the major trend clearly follows the revised scaling f_scal^∗ according to eq. (4.3). Merely some I-phases at H-L transitions dier a little from the main trend. Also, I-phases in Hydrogen plasmas in unfavourable USN conguration do not t into the adapted scaling. However, with the original scalingf_scal, they would t in perfectly.

4.7 Further Observations

4.7.1 Frequency-Amplitude Correlation

As suggested by g. 4.9 and 4.10, the frequency and the amplitude of the peaks of the limit-cycle oscillations are to some extent correlated. When the frequency decreases during the I-phase, the amplitude of the peaks ofB˙ increases at the same time. For both illustrations, the corresponding frequencies for the peaks were determined with the ∆t method. Only the rst 30 peaks in each I-phase were considered in order to keep the plots as clear as possible.

In g. 4.9, the discharge with the clearest correlation between the frequency and the amplitude (#29306) is depicted. Apart from two outliers on the right, the data points

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1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 f (kHz)

2 3 4 5 6 7 8

_B (T/s)

#29306

Figure 4.9: Frequency- amplitude correlation for discharge #29306 along with a least-squares t. Except for the two outliers on the right, the data points lie on one straight line.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 f (kHz)

2 3 4 5 6 7 8

_B (T/s)

29303 29306 29308 29310

Figure 4.10: Frequency- amplitude correlation for a set of 4 selected I-phases along with their least- squares ts. For each discharge considered, a linear dependence of the amplitude on the frequency is indicated. An overall trend, however, can not clearly be identied.

make up a nice decreasing line reaching from 1.3 to 2.1 kHz and from 8 to 2.5Ts⁻¹. The red line illustrates the least-squares t for this data set.

Fig. 4.10 displays a collection of 4 selected I-phases featuring a relatively strong correlation along with their least-squares ts. For each individual discharge, a linear dependence of the amplitude on the frequency is discernible. But the resulting linear relations have dierent slopes and a major trend can not be recognised. This impression intensies when even more discharges are regarded. The plot gets even more scattered and a strict coherence between the amplitude and the frequency can not be conrmed.

The amplitude of B˙ in the I-phase appears to be also depending on other plasma parameters or quantities besides the frequency of the LCOs. Thus, an elaborate study of the amplitude dependencies is advisable.

4.7.2 Indication of a Kink in the Frequency Dependence

In some discharges, a kink in the frequency scaling occurred if the rst 60 peaks were considered. In others, more peaks, for example 200, had to be considered in order to

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1.0 1.5 2.0 2.5

f (kHz) 1.0

1.5 2.0 2.5

fscal (kHz)

#29306

Figure 4.11: Frequency scaling for discharge #29306 calculated with the ∆t method for the rst 60 peaks. The scaling f_scal according to eq. (4.2) seems to hold true only for frequencies higher than 1.4 kHz in this discharge. For frequencies below the kink, indicated by the blue circle,f_scalappears to be constant and clearly deviates from the bisectrix.

identify the kink. An example (discharge #29306) is depicted in g. 4.11. In this case, the frequency has been determined with the∆tmethod and 60 peaks were considered. Below a frequency of about 1.4 kHz, the data points form a horizontal line wheref_scaldoes not change. For higher frequencies, the predicted frequencyf_scal and the actual frequencyf coincide very well. This behaviour occurred in several I-phases in the reference discharges.

Fig. 4.12 displays the situation when all reference discharges are considered. In this case, the frequency was determined via the spectrogram method. Fig. 4.12a shows the frequency scaling for the rst 60 peaks in all I-phases. No kink is visible in this instance.

In g. 4.12b, the rst 200 peaks in all I-phases are considered. As a result, a kink may be identied at about 1.3 kHz. This might be an indication of the plasma entering another regime below a frequency of 1.3 kHz, in which the frequency dependence on n,T and B˙ according tof_scal does not apply.

In summary, the scaling according tof_scal seems to pertain only for frequencies higher

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (kHz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

(a) Frequency scaling for 60 peaks.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (kHz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fscal (kHz)

(b) Frequency scaling for 200 peaks.

Figure 4.12: Frequency scaling for all reference discharges. For the rst 60 peaks, depicted in (a), no kink in the scaling is visible. If one considers more peaks, as shown in (b), a kink in the scaling might be identied at a frequency of about 1.3 kHz (blue circle).

than 1.3 kHz but for the whole data set, the kink is dicult to identify. Although the shift in the frequency dependence is visible for some discharges, it cannot be considered as a general feature pointing to a second regime of LCOs with a dierent physical origin of the dynamics.

4.8 Heating Power Dependence

The occurrence of I-phases depends on the heating power and the density in the plasma [14]. These dependencies are inuenced by the type of the plasma and its conguration.

The net heating power Pnet is the overall heating power, adjusted by losses and absorp- tion coecients of the respective heating methods, minus the variation of the plasma energy [24]. The power threshold denes the heating power that is needed to trigger the transition into the H-mode for a given density.

Fig. 4.13 illustrates, where the I-phases in the dierent congurations occur depending on the heating power P_net and the density n¯_e,centre. The grey line indicates the power threshold above which the plasma is expected to enter the H-mode for IP = 1 MA and Bt = 2.35 T [24]. In this plot, an I-phase from discharge #27126 (L-H transition, Deuterium plasma and normal conguration) has been added to the reference (red).

The I-phases of the reference discharges emerge at comparative low densities of about 2.5 to 4·10⁻¹⁹m⁻³ and a heating power between 1 and 2 MW. This conrms what has already been found in [14]. The same holds true for the discharges in USN conguration.

Again, this is exactly what one would expect since only the shape of the plasma and the

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magnetic eld are inverted. This is another proof of the accordance of plasmas in upper and lower single null conguration.

The one protruding I-phase at extremely high density and heating power originates from discharge #30863, which was the only discharge in unfavourable USN conguration.

This implies, that the inversion of the magnetic eld in USN congurations has a strong impact on the density and heating power that is needed to enter the I-phase. Nevertheless, even this I-phase appears close to the L-H power threshold.

In Helium plasmas, the I-phase seems to occur at higher densities compared to the reference discharges. This is remarkable since I-phases at L-H transitions in Deuterium do practically not appear aboven¯_e= 5·10¹⁹m⁻³. I-phases in Hydrogen plasmas appear at higher heating power. This might be related to the fact that the heating power thresholdP_thres in Hydrogen is much higher than in Deuterium [24].

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1 2 3 4 5 6 7 8 9 10

¹

n_e;centre (10^¡¹⁹m^¡³)

0 1 2 3 4 5 6 7

Pnet (MW)

Reference HeH

Unfavourable USN USNPower Threshold

Figure 4.13: Net heating power Pnet of the plasma and density n¯e,centre at which the I-phases in dierent congurations and types of plasmas occur. One can see that the discharges in USN conguration enter the I-phase at similar values for Pnet and

¯

ne,centreas the ones in normal conguration (Reference). The discharge in unfavourable USN conguration (#30863) stands out, as its I-phase takes place at extremely high density and heating power but still seems to aliate to the heating power threshold.

Compared to the reference, I-phases in Hydrogen plasmas occur at higher heating power and Helium plasmas seem to enter the I-phase at higher densities.

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Chapter 5 Conclusion

The aim of this thesis was to investigate the frequency scaling of limit-cycle oscillations in dierent types and congurations of fusion plasmas. In the course of the investigations, two dierent methods for the frequency determination, the ∆tmethod and the spectrogram method, have been elaborated and employed. It was shown that both methods practically yield the same results and both of them have been applied according to their specic advantages.

The formula for the frequency scaling has been revised in order to also apply for Helium and Hydrogen plasmas as well as dierent plasma congurations. The outcome of this revision was, that the frequency of the limit-cycle oscillations during I-phases in fusion plasmas scales as

f_scal^∗ = 1.542·10⁶· n¯^−1.10_e,centre T_e^−0.91 B_t^1.93 q₉₅^−1.36 ·q²

m (5.1)

wheref_scal^∗ [Hz] is the resulting frequency,n¯e,centre [10¹⁹ m⁻³] the line averaged electron core density ,T_e[eV] the electron temperature,B_t[T] the toroidal magnetic eld andq₉₅ the safety factor. q is the charge, given in units of the elementary charge [e], and mthe mass, given in atomic mass units [u], of the nucleus of the element or isotope of which the plasma consists of. This formula is a generalisation of that proposed by G. Birkenmeier et al. in [4]. The factorq²/m, which I have introduced to the scaling, ensures that the scaling also covers Hydrogen and Helium plasmas.

In this thesis it could be shown that this formula applies for normal LSN, USN and unfavourable USN conguration, for Helium and Hydrogen plasmas and partially also for H-L transitions. In order to approve the latter, more research should be conducted.

Further, the scaling could not be conrmed for Hydrogen plasmas in unfavourable USN conguration. Since the respective discharges were very specic, as they featured multiple simultaneous modications of the plasma type and conguration, this does not prove the scaling wrong. On the contrary, it is remarkable that, apart from these exotic discharges, the scaling given by eq. (5.1) holds true for various plasma types and congurations and seems to be universally applicable.

Furthermore, the heating power dependence of the transition into H-mode for LSN and USN discharges as well as Helium and Hydrogen plasmas has been validated. It is found

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that the I-phase occurs close to the L-H power threshold in all considered plasma types and congurations even for extreme values of density and heating power.

In addition, a number of further observation were made. On the one hand, a correlation between the frequency and the amplitude of the LCOs has been observed but a general linear dependence could not be conrmed. A more detailed examination should be conducted in order to ascertain the correlation between the frequency and the amplitude and possibly other quantities. On the other hand, a kink in the frequency scaling of the LCOs was indicated. The dependence of the frequency does not seem to hold true for small frequencies below 1.3 kHz. However, further investigations regarding this occurrence are advisable in order to clarify whether there is a change in frequency dependence and under which circumstances the kink appears.

In summary, the adapted scaling according to eq. (5.1) seems to apply very well for various plasma types and congurations and, apart from scenarios with multiple simultaneous modications, has proven itself very robust. Thus, a theoretical description of the LCOs has to agree with this frequency scaling.

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Frequency Scaling of Limit-Cycle Oscillations