Stability and properties of electron-driven fishbones in tokamaks

 

Stability and properties of electron-driven fishbones in tokamaks

 

Thèse de doctorat

soutenue le 29/11/2012 par

Antoine Merle

en vue d’obtenir le grade de

Docteur de l’École Polytechnique

Spécialité : Physique

Responsable CEA Joan Decker Ingénieur de recherches CEA Directeur de thèse Xavier Garbet Directeur de recherches CEA Rapporteur Jonathan Graves Professeur à l’EPFL Rapporteur Fulvio Zonca Professeur à l’ENEA Examinateur Sadruddin Benkadda Directeur de recherches CNRS Examinateur Jean-Marcel Rax Professeur à l’École Polytechnique

Abstract

In tokamaks, the stability of magneto-hydrodynamic modes can be modified by populations of energetic particles. In ITER-type fusion reactors, such populations can be generated by fusion reactions or auxiliary heating. The electron-driven fishbone mode belongs to this category of instabilities. It results from the resonant interaction of the internal kink mode with the slow toroidal precessional motion of energetic electrons and is frequently observed in present-day tokamaks with Electron Cyclotron Resonance Heating or Lower Hybrid Current Drive. These modes provide a good test bed for the linear theory of fast-particle driven instabilities as they exhibit a very high sensitivity to the details of both the equilibrium and the electronic distribution function.

In Tore Supra, electron-driven fishbones are observed during LHCD-powered discharges in which a high-energy tail of the electronic distribution function is created. Although the destabilization of those modes is related to the existence of a fast particle population, the modes are observed at a frequency that is lower than expected. Indeed, the corresponding energy assuming resonance with the toroidal precession frequency of barely trapped electrons falls in the thermal range.

The linear stability analysis of electron-driven fishbone modes is the main focus of this thesis. The fishbone dispersion relation is derived in a form that accounts for the contribution of the parallel motion of passing particles to the resonance condition. The MIKE code is developed to compute and solve the dispersion relation of electron-driven fishbones. The code is successfully benchmarked against theory using simple analytical distributions. When coupled to the relativistic Fokker-Planck code LUKE and to the integrated modeling platform CRONOS, it is used to compute the stability of electron-driven fishbones using reconstructed data from tokamak experiments. Using the code MIKE with parametric distributions and equilibria, we show that both barely trapped and barely passing electrons resonate with the mode and can drive it unstable. More deeply trapped and passing electrons have a non-resonant effect on the mode that is, respectively, stabilizing and destabilizing. MIKE simulations using complete ECRH-like distribution functions show that energetic barely passing electrons can contribute to drive a mode unstable at a relatively low frequency. This observation could provide some insight to the understanding of Tore Supra experiments.

Résumé

La stabilité des modes magnéto-hydrodynamiques dans les plasmas de tokamaks est modifiée par la présence de particules rapides. Dans un tokamak tel qu’ITER ces particules rapides peuvent être soit les particules alpha créées par les réactions de fusion, soit les ions et électrons accélérés par les dispositifs de chauffage additionnel et de génération de courant. Les modes appelés fishbones électroniques correspondent à la déstabilisation du mode de kink interne due à la résonance avec le lent mouvement de précession toroidale des électrons rapides. Ces modes sont fréquemment observés dans les plasmas des tokamaks actuels en présence de chauffage par onde cyclotronique électronique (ECRH) ou de génération de courant par onde hybride basse (LHCD). La stabilité de ces modes est particulièrement sensible aux détails de la fonction de distribution électronique et du facteur de sécurité, ce qui fait des fishbones électroniques un excellent candidat pour tester la théorie linéaire des instabilités liées aux particules rapides.

Dans le tokamak Tore Supra, des fishbones électroniques sont couramment observés lors de décharges où l’utilisation de l’onde hybride basse crée une importante queue de particules rapides dans la fonction de distribution électronique. Bien que ces modes soit clairement liés à la présence de particules rapides, la fréquence observée de ces modes est plus basse que celle prévue par la théorie. En effet, si on estime l’énergie des électrons résonants en faisant correspondre la fréquence du mode avec la fréquence de précession toroidale des électrons faiblement piégés, on obtient une valeur comparable à celle des électrons thermiques.

L’objet principal de cette thèse est l’analyse linéaire de la stabilité des fishbones électroniques. La relation de dispersion de ces modes est dérivée et la forme obtenue prend en compte, dans la condition de résonance, la contribution du mouvement parallèle des particules passantes. Cette relation de dispersion est implémentée dans le code MIKE qui est ensuite testé avec succés en utilisant des fonctions de distributions analytiques. En le couplant au code Fokker-Planck relativiste LUKE et à la plate-forme de simulation intégrée CRONOS, MIKE peut estimer la stabilité des fishbones électroniques en utilisant les données reconstruites de l’expérience. En utilisant des fonctions de distributions et des équilibres analytiques dans le code MIKE nous montrons que les électrons faiblement piégés ou faiblement passants peuvent déstabiliser le mode de kink interne en résonant avec lui. Si l’on s’éloigne de la frontière entre électrons passants et piégés, les effets résonants s’affaiblissent. Cependant les électrons passants conservent une influence déstabilisante alors que les électrons piégés tendent à stabiliser le mode. D’autres simulations avec MIKE, utilisant cette fois des distributions complètes similaires à celles obtenues en présence de chauffage de type ECRH, montrent que l’interaction avec les électrons faiblement passants peut entraîner une déstabilisation du mode à une fréquence relativement basse ce qui pourrait permettre d’expliquer les observations sur le tokamak Tore Supra.

Contents
List of Figures

Chapter 1 Introduction

1.1 Nuclear fusion

In a nuclear fusion reaction, two light nuclei are brought together to form one heavier element. If the mass of the products of the reaction is smaller than the total mass of the initial elements, the reaction releases energy. The source of this energy is the strong nuclear interaction which binds the protons and neutrons inside the nucleus. This process of nuclear fusion is very efficient in terms of energy production per mass of the reactants, far above processes involving chemical reaction like the oil combustion. But this tremendous energy comes at a price, indeed in order to fuse the reactants must overcome their mutual repulsion due to the Coulomb interaction between the two positively charged nuclei.

Today nuclear fusion is studied as a potential energy source. The most accessible reaction is the one involving deuterium and tritium , two heavy isotopes of hydrogen, and producing one Helium nucleus (also named -particle) and a neutron ,

(1.1)

The numbers between parenthesis are the amount of kinetic energy carried by the fusion products, such that the total energy released per reaction is . The temperature of the reactants plays an important role in reaching an efficient energy production, since they must carry enough kinetic energy to overcome the Coulomb barrier. The reaction rate reaches a maximum when the thermal energy is about (). At this level the deuterium and the tritium form a fully ionized gas or plasma.

Until the plasma can be self-heated by fusion reactions, one has to inject energy into the plasma to bring and maintain the fuel at the required temperature due to energy losses. The Lawson criterion [law57] states that the fusion power overcomes the power losses when the product reaches a certain value, where is the fuel density and is named the energy confinement time and is defined in a steady-state regime as the ratio of the energy content of the plasma and the level of power losses . At the product must reach the value of . Two different approaches can be considered to satisfy this criterion.

  • Achieve a very high density plasma () for a short time (). In inertial confinement devices, these conditions are achieved by compressing D-T targets with powerful lasers.

  • Maintain a low density plasma () for a longer time (). In magnetic confinement devices, the plasma is confined by a strong magnetic field which keeps the plasma from cooling down on the wall of the reactor.

1.2 Magnetic confinement fusion

Charged particles in a magnetic field follow trajectories which are helically wound around magnetic field lines. The extent of the helix perpendicular to the magnetic field line is called the Larmor radius or gyration radius and is inversely proportional to the amplitude of the magnetic field. For a particle of mass and charge and with a velocity in the direction perpendicular to the magnetic fiel of amplitude , the Larmor radius is

(1.2)

Thus a stronger magnetic field will provide better confinement properties. In present day magnetic confinement machines, the magnetic field amplitude is typically of several teslas (), while the earth magnetic field has an amplitude of a few . The parameter measures the ratio of the plasma kinetic energy and the magnetic energy

(1.3)

in magnetic confinement devices is generally of the order of .

The confinement properties depend also on the geometry of the magnetic field. Initially linear devices with open field lines were tested but the energy confinement times measured were not compatible with a sustainable production of energy due to the important particle and energy losses at both ends of the machines. The simplest configuration with closed magnetic field lines (or at least closed magnetic surfaces) is when the magnetic field lines form a torus. Unfortunately if the magnetic field is purely toroidal, the charged particles suffer a vertical drift due to the curvature of the magnetic field lines and are not confined. But if one adds a poloidal component to the magnetic field so as to make the field lines wind helically around the torus, then the particles orbits are periodic and are confined to the reactor chamber.

1.3 The tokamak configuration

The tokamak is currently the most successful configuration based on this idea. In this configuration, the toroidal field is produced by vertical coils surrounding the torus while the poloidal magnetic field is produced by an intense toroidal electric current which flows inside the torus. The magnetic system of a tokamak is presented in figure 1.1, where additional coils needed for the plasma shape and stability control have been added.

Figure 1.1: The tokamak configuration coil system.

The level of performance of the plasma can be measured by introducing the enhancement factor which corresponds to the ratio of the power released into the plasma by fusion reactions and the level of power injected into the plasma . The limit is called the break-even, it corresponds to the state where the plasma is sustained to equal parts by the fusion energy power and by the external power input. This has been achieved in the JET (for Joint European Torus) tokamak [jac99]. The ITER tokamak actually in construction is designed to operate routinely at when operating with D-T fuel. The predicted fusion power output is of the order of well above the present record of with the JET tokamak, this level should be maintained for as long as . The characteristics of the ITER machine can be found in table 1.1 along with those of the JET and of the Tore Supra tokamak.

Tore Supra JET ITER
Major radius ()
Minor radius ()
Plasma volume ()
Plasma current ()
Magnetic field amplitude ()
Pulse Duration ()
Fuel mix D-D D-D / D-T D-T
Fusion Power ()
Amplification factor
Table 1.1: Principal parameters for the Tore Supra, JET and ITER tokamaks.

A commercial fusion reactor should operate around while the limit is called ignition.

The large plasma current necessary for the plasma stability is usually induced by a secondary set of electromagnets which create an inductive toroidal electric field inside the plasma which in turn creates an electric current due to the finite resistivity of the plasma. Simultaneously the plasma is heated by Joule effect, this process is the principal source of plasma heating and current drive in most present day machines. But at high temperatures the resistivity and the efficiency of the Joule heating drop and additional heating techniques have been developed.

  • The Neutral Beam Injection (NBI) system: since charged particles cannot enter the plasma due to the magnetic field, deuterium ions are accelerated to an energy of about before being neutralized. Once inside the plasma the atoms are stripped from their electrons. The energy of the energetic ions is then transferred to the background plasma by successive collisions.

  • Ion Cyclotron Resonance Heating (ICRH): electromagnetic waves are sent into the plasma at the ion cyclotron frequency. The resonant interaction between the particles and the waves results in a net transfer of energy from the waves to the particles and therefore heats the plasma. The ion cyclotron frequency is is of the order of and lies in the radio-frequency part of the electromagnetic spectrum. Note that in a tokamak, the magnetic field amplitude is typically inversely proportional to the major radius such that the region where the particles can resonate with the wave is limited to a layer near a given value of the major radius.

  • Electron Cyclotron Resonance Heating (ECRH) follows the same principle as ICRH. The frequency of the waves matches the electron cyclotron frequency which is of the order of (microwaves).

For the ITER tokamak the amount of power available is of about , of deuterium neutral beams and of radio-frequency heating [iterweb].

When aiming at long discharges the problem of maintaining the plasma current for a long period of time arises. Since the amount of flux which can be varied through the secondary circuit formed by the plasma is finite, the plasma current cannot be sustained solely by the transformer for an infinite time. In ITER an important part of the total plasma current will be driven non-inductively (not relying on the transformer). Non-inductive current-drive can be achieved by

  • Radiofrequency waves. This technique relies once again on the resonant absorption of electron-magnetic waves. The spatial structure of the waves generates an electric field which accelerates the electrons primarily in the parallel direction. Currently two different techniques have been used, the first one uses the electron cyclotron resonance and is called Electron Cyclotron Current-Drive or ECCD. The second one uses the lower-hybrid resonance, one then speaks of Lower-Hybrid Current-Drive or LHCD. The efficiency of the current-drive is measured by the ratio of the driven current and the amount of power injected by the waves. Typical values for the efficiency are around .

  • Bootstrap current. Due to the inhomogeneity of the magnetic field, some particles are trapped in the region of low magnetic field (or low field side noted LFS, the region of high magnetic field on the inboard side is called the high field side and is noted HFS). When the collision frequency is lower than the bounce frequency (the orbit frequency of trapped particles) ,the existence of these trapped particles and of a radial pressure gradient produces a parallel current called the bootstrap current. This current is important in the regions of strong pressure gradients such as transport barriers. During the ITER tokamak operation, the level of bootstrap current is expected to be around .

The good confinement properties of the plasma are guarantied by the existence of nested flux-surfaces which means that surfaces exist such that the magnetic field is everywhere tangent to those surfaces. Let be a flux-label, we define the safety factor by

(1.4)

where and are the contravariant components of the magnetic field in the toroidal and poloidal directions. The value of corresponds to the average field-line pitch and can be interpreted as the number of toroidal turns done by a field line for every poloidal turn. Depending on the value of two type of surfaces exist.

  • If is an irrational number then each field line on this surface fills the surface ergodically. Due to the large parallel heat conductivity of electrons, the pressure is homogenized over the whole flux-surface and is a quantity which depends only on the flux label .

  • If is a rational number then the field lines are closed. These surfaces are prone to instabilities since perturbations with structures aligned with the magnetic field and minimize the bending of the magnetic field lines will grow more easily. If these structures have the following spatial dependence .

In this thesis, one of these instabilities is studied, the electron-driven fishbone mode.

1.4 Introduction to the electron-driven fishbone mode

The electron-driven fishbone mode belongs to the so-called category of energetic particle driven instabilities. Energetic particles correspond to particles with a velocity higher than the thermal velocity . In tokamaks the additional heating and current-drive systems such as NBI, ICRH, ECRH/ECCD or LHCD provide large sources of energetic ions and electrons such that the population of those particles is generally higher than the level in a purely maxwellian distribution. The fusion reactions produce -particles at an energy of and are therefore another source of energetic particles. The characteristic frequencies of the motion of those particles, in particular the slow toroidal precession motion due to the magnetic drifts, are in the same range as the frequencies of the Magneto-HydroDynamic (MHD) instabilities allowing a resonant interaction between the particles and the waves. These instabilities have a radial extent corresponding to a large fraction of the minor radius and are well described by the MHD model which considers the plasma as a magnetized fluid.

1.4.1 The fishbone instability

The fishbone instability was first observed in the PDX tokamak during high- experiments using NBI in near-perpendicular injection, meaning that the initial velocity of the injected ions is nearly perpendicular to the magnetic field lines [mcg83].

Figure 1.2 reproduces the original figure describing the fishbone instability from reference [mcg83].

Figure 1.2: Original report of the fishbone instability with traces of the soft X-ray emissivity (top), of the poloidal magnetic field variations (middle) and neutron emissivity (bottom). Taken from [mcg83].

The instability appears at high levels of neutral beam power in the form of successive bursts of activity on the central soft X-ray signals and Mirnov coils measurements (see the top and middle panels of figure 1.2). The instability is located in the plasma core . The name fishbone was given because of the shape of the signal on the Mirnov coils. The activity was correlated with a drop in the measured neutron emissivity (see the bottom panel of figure 1.2) indicating a loss of the energetic ion content of the plasma. This was correlated with measures of the energetic ion distribution with the charge-exchange diagnostic indicating that the population of ions with energies between and (where is the injection energy of the beam ions) drops immediately after the bursts. The measured frequency of the instability was in the ion diamagnetic direction and was compatible with the precession drift frequency of deeply trapped ions with energies close to which are abundantly produced by near-perpendicular beam injection. It was then suggested that both the energetic ion losses and the mechanism of the instability growth were linked to the resonant interaction of the particles with the mode.

Later, ion-driven fishbone instabilities were reported on other machines such as TFTR [kai90], JET [nav91], JT-60 [nin88] or DIII-D [hei90]. The instabilities were observed during NBI heating with or without additional ICRF heating, the measured frequencies were situated close to the precession frequency of energetic ions or to the ion diamagnetic frequency or in-between those two frequencies.

1.4.2 Theoretical interpretation

Two theoretical models were proposed to interpret these instabilities [che84, cop86]. Both rely on the modification of the ideal stability of the internal kink by resonance with a population of energetic ions. The resonance occurred at the precession drift frequency of trapped ions. The source of the instability is the radial gradient of the distribution function of ions. A negative radial gradient, which corresponds to a central deposition of beam ions is necessary for the growth of the instability. The difference between the two models being that; in the model proposed by Chen et al. [che84] the frequency is fixed by the precession frequency of deeply trapped ions such that the mode is a continuum resonant mode, while in the Model of Coppi et al. [cop86] the frequency is close to the ion diamagnetic frequency such that the mode is described as a discrete gap mode. In fact these 2 models can be described using a single formalism [big87].

1.4.3 Sawtooth stabilization

This formalism was also used to explain the stabilization of the sawtooth instabilities by energetic ions and the apparition of monster sawteeth such as the ones observed on the JET tokamak [cam88]. An example of a monster sawtooth is shown on figure 1.3.

Figure 1.3: A characteristic monster sawtooth discharge in JET during ICRF minority heating. is the central electron temperature, is the plasma stored energy, is the D-D fusion reaction rate, is the line-averaged electronic density and is the level of ICRH power [bha89].

The sawtooth instability is a periodic relaxation of the plasma core which is constituted of a ramp-up phase where the core plasma temperature rises followed by the apparition of an precursor and by a sudden crash of the temperature profile over the whole central region where . The precursor has the same structure as the one of the fishbone mode, which is the one of the internal kink mode.

In the JET experiments, the sawtooth instability is stabilized by an input of ICRH power and the temperature crash can be triggered by cutting the ICRH power, in this case the crash occurs to after the end of the RF pulse which is consistent with the slowing down time of the energetic ions. This confirms the assumption of a stabilization by energetic particles.

White et al. showed, using analytical trapped fast-ion distributions, that a window of values for the fast-ion beta parameter existed in which both the sawtooth and the fishbone instability were stable [whi89].

1.4.4 Electron-driven fishbones

The initial theory of Chen et al. only considered the modification of the internal kink stability by energetic trapped ions. But it was later showed [sun05, wan06a, zon07, wan07] that it could be applied to the influence of energetic electrons since the precession frequency has the same absolute value for ions and electrons at the same energy. Yet the drift motion of electrons has to be reversed and the distribution function has to have an inverted radial gradient for a transfer of energy from the electrons to the mode, due to the fact that the internal kink mode rotates preferably in the ion diamagnetic direction. In this case the resonant drive is mostly provided by barely trapped electrons.

Electron-driven fishbones in DIII-D

The first report of electron-driven fishbones was published by Wong et al. [won00] for the DIII-D tokamak. The fishbones were observed in discharges where off-axis ECCD was used to obtain negative magnetic shear (a region where decreases with radius) in the central region. Bursts of fishbone activity appeared when the ECCD power was deposited just outside the surface. The inversion of the radial gradient of energetic barely trapped electrons was confirmed by numerical reconstruction of the electronic distribution function as can be seen on figure 1.4.

Figure 1.4: Spatial profile of the population of electrons at the trapped-passing boundary for specified energies. The data comes from the reconstructed suprathermal electron distribution in the DIII-D shot 96163 [won00].

The influence of barely trapped electrons was also confirmed by varying the poloidal angle of the position of the peak in power deposition (but keeping its radial position outside of the surface), the fishbone activity was maximum when the power deposition peaked on the inboard midplane corresponding to optimal conditions for the production of energetic barely trapped electrons. It should be noted that energetic ions were present in the plasma due to NBI heating but their influence was ruled out by the authors.

Electron-driven fishbones in FTU

Fishbone instabilities driven by suprathermal electrons have also been observed in FTU using LHCD only [rom03, zon07, ces09]. In FTU two different regimes were obtained as can be seen on figure 1.5.

Figure 1.5: Electron fishbone observation in FTU, is the line-averaged density, is the level of LHCD power. The two bottom panels use the ECE radiometer data and show the fast electron temperature fluctuations and the central radiation temperature respectively [zon07].

At a moderate level of LHCD power the growth of an instability is observed on ECE radiation fluctuation measurements until this instability reaches a saturated level. A simultaneous diminution of the central radiation temperature is observed indicating the loss of energetic electrons. At higher levels of LHCD power the typical bursts of activity appear in conjunction with drops of the central radiation temperature similar to the drops in neutron rate measurements in the case of ion-driven fishbones. Using a linear stability analysis [zon07], it has been established that in the case of the saturated mode the fast particle beta is just above marginal stability whereas it is well above marginal stability in the bursting regime.

Electron-driven fishbones in Tore Supra

Electron-driven fishbones are also observed on the Tore Supra tokamak during LHCD discharges [gon08, mac09]. The modes are observed during the so-called oscillating regime or O-regime where the equilibrium profiles such as and experience periodic oscillations [gir03, imb06], slow frequency chirping but also frequency jumps are observed corresponding to a modification of the structure of the mode (modification of and ). The modes are also seen during steady-state discharges with fixed equilibrium profiles, the modes frequency and structure is similar to those occurring in oscillating discharges. More recently similar modes were observed in-between sawteeth.

(a)
(b)
Figure 1.6: Central temperature evolution (top a and b), spectrogram of the ECE radiation fluctuations (bottom a) and energy of resonant barely trapped electrons (bottom b) for the Tore Supra discharge #41117. The black line with squares corresponds to a mode, the magenta line with circles to a mode, the red line with triangles corresponds to a mode and the green line with stars to a mode [gui12].

Figure 1.5(a) reproduces the central electron temperature evolution of the Tore Supra discharge number 41117 together with the spectrogram of one of the central ECE channels. If one considers the time where the central electron temperature is maximum as a reference, the frequency decreases at each frequency jump, beginning around down to then and finally . Each frequency jumps is correlated with a modification of the structure of the mode, the analysis of the radial structures taking into account the vertical extent of the ECE antenna and the alignment of the antenna line of sight with the plasma midplane showed that the poloidal mode numbers are successively , , and [gui12]. Since the radial position of the modes is consistent with the position of the surface of the reconstructed equilibrium, the toroidal mode numbers are assumed to match the poloidal mode numbers. While the modes have relatively constant radial positions during the cycle, the one of the mode drifts slowly inward [gui11]. This observation, together with the fact that both and modes are present at the same time with the mode being located further from the plasma center, indicates that the magnetic shear is low inside the surface. Moreover low-shear profiles are known to be more unstable to modes with high mode numbers [wes86, has88].

In order to estimate the energy of the resonant electrons, the Doppler shift due to the plasma toroidal rotation has to be estimated. This evaluation was made using an diamagnetic mode which does not appear on the spectrogram of figure 1.5(a), the measured frequency of this mode is around while the ion-diamagnetic frequency in TS is typically of a few hundred kHz. The mode measured frequency was therefore assumed to be dominated by plasma rotation and the value was retained [gui12]. The energy of resonant barely trapped electrons was then estimated by matching the frequency of the modes in the plasma rest-frame with the precession frequency of barely trapped electrons , the results are shown in figure 1.5(b). It appears that this energy is comparable to the energy of thermal particles around .

Electron-driven fishbones in other tokamaks

Other machines, such as COMPASS-D [val00], HL-1M [din02] or HL-2A [che09] also reported observations of electron-driven fishbone modes. In all cases the frequency of the mode is observed to be in or near the ion diamagnetic gap of the Alfvén spectrum except in the case of COMPASS-D where the frequency is higher () and close to the TAE frequency.

It should be noted that the theory also allows the existence of fishbones rotating in the electron direction and driven by deeply trapped electrons but these would be more heavily damped by coupling to the MHD continuum and would therefore require a stronger drive than in the case of electron-driven fishbones rotating in the ion direction [zon07]. Some numerical simulations were able to produce such instabilities [vla11, vla12].

1.5 Thesis motivation and outline

The example of the fishbone instability described in the previous section shows that populations of energetic particles can give rise to macro-scale instabilities through resonant interaction. Simultaneously this interaction affects the confinement of the particles. On the one hand the development of such instabilities could prevent the fusion-born alpha particles from transferring their energy to the plasma bulk [hei94, ITE99, fas07]. On the other hand this phenomenon is considered as a way to control the accumulation of the helium ash made up of the slowed-down alpha particles which can affect the fusion reaction rate by diluting the fuel [ITE99]. Energetic-particle driven instabilities can also affect the power deposition profiles of auxiliary heating systems by modifying the spatial distribution of the populations of energetic particles, one corollary being that we can use this phenomenon to control these profiles. Whether to prevent anomalous energetic particle transport or to provide new control mechanisms for the plasma, it is important to understand the mechanisms of the onset of such instabilities.

The study of electron-driven fishbones is directly relevant to the study of the interaction of alpha particles with low frequency MHD instabilities since in this case the resonance would happen at the toroidal precession frequency of energetic particles which depends on the energy of the particles and not their mass. Also energetic electrons have very thin orbits much like fusion-born alphas in ITER [zon07]. Moreover the stability of electron-driven fishbones is very sensitive to the details of both the electronic distribution function and the safety factor profile [dec09a, mer10]. Thus they provide a sensitive test for the linear stability model.

In the Tore Supra tokamak electron-driven fishbones have been observed at a frequency well below the precession frequency of barely trapped energetic electrons which was the one predicted by the theory. The aim of this thesis is to study the stability of electron-driven fishbones to provide a possible explanation for this phenomenon.

In the first three chapters we introduce some of the tools necessary to our analysis. Chapter 2 is dedicated to the description of the equilibrium magnetic field configuration in a tokamak, the formalism developed is then used in chapter 3 where a hamiltonian formalism is used to study the motion of the particles in a tokamak. In chapter 4 we introduce the framework of the ideal MHD energy principle which is used to study MHD instabilities in magnetized plasmas. The stability of the internal kink mode is investigated in chapter 5 using this formalism. The final part of this thesis is dedicated to electron-driven fishbones. The modification of the internal kink dispersion relation by resonance with energetic particles is then derived in 6 using a kinetic description of energetic electrons. Special care is given to the resonance with passing particles which are of importance in the case of energetic electrons. The MIKE code which implements this model is introduced in chapter 7 and is used in chapter 8 where we show that the resonance with passing electrons lowers not only the density of energetic electrons at the instability threshold but also the frequency of the mode.

Chapter 2 Magnetic configuration

The configuration of the magnetic field in a tokamak is investigated. In the first section different coordinate systems used to describe the magnetic field are introduced. In the second section the Grad-Shafranov equation [gra58, sha58] which describes the equilibrium configurations for a toroidal magnetic field is derived. Finally we introduce some notations which will be used throughout this thesis.

2.1 Coordinate system

The case of an axisymmetric magnetic field with nested flux-surfaces is considered. The innermost flux-surface is degenerate and is called the magnetic axis. The axis of symmetry is supposed to be in the vertical direction . Three different coordinate systems are defined:

  • a right-handed orthonormal Cartesian coordinate system ,

  • a right-handed polar coordinate system such that ( will be further referred to as the geometrical toroidal angle) and is the distance to the vertical axis,

  • a general coordinate system where is a flux-label ( is constant on a given flux-surface and its value on the magnetic axis is chosen to be ) is a poloidal angle such that corresponds to the outboard midplane and a toroidal angle.

The standard definitions for the covariant basis are used

(2.1)

the contravariant basis

(2.2)

the metric tensor elements

(2.3)

and the jacobian such that

(2.4)

The following identities hold

(2.5)
(2.6)
(2.7)

where is the Kronecker symbol and is the antisymmetric tensor.

Because an axisymmetric field is considered, one can choose to be the geometric toroidal angle and such that is orthogonal to both and . In this way, one has

2.2 Vector potential and magnetic field

The vector potential of an axisymmetric magnetic field can be put in the form (see [whi03])

(2.8)

is, up to a constant, the flux of the magnetic field through a poloidal surface (which is defined by , and an arbitrary constant).

In the same way is times the flux of the magnetic field though a toroidal surface (which is defined by , and an arbitrary constant).

Without loss of generality, one can choose which is a flux-label as the radial variable. The derivative of against is known as the safety factor and is denoted . It is a function of only. Its inverse is known as the rotational transform. The contravariant representation of the magnetic field is then easily obtained,

(2.9)
(2.10)
(2.11)

where is the jacobian of the coordinate system, such that .

2.3 Magnetic field lines

Consider a magnetic field line , the magnetic field is aligned with the tangent of the field line for all :

(2.12)

Solving this ordinary differential equation, one obtains first that is constant along the field line, which is not surprising since magnetic field lines are embedded in magnetic flux surfaces, but one obtains also the relationship between and along the field line,

(2.13)

such that

(2.14)

The physical meaning of is now apparent. If , then the pitch of the field lines in the plane, , is a function of alone and the field lines are straight. Now, if , the pitch of the field-lines is not constant anymore, but its average on one poloidal period is still .

A set of coordinates with is called flux-coordinates or straight field line coordinates. It can be shown that for well-behaved fields, such coordinates always exist, and that they can be found starting from any set of coordinates where is a flux label, is a poloidal angle and a toroidal angle, by only modifying either the poloidal angle or the toroidal angle.

A method to obtain flux coordinates by modifying only the poloidal angle is presented. Let be the new poloidal angle. Without loss of generality can be written where is a periodic function of . Then from the contravariant form of , one has

(2.15)
(2.16)

from which a solution is . 111Similarly, is a new toroidal angle such that are flux-coordinates.

Then is a measure of the distance between the actual coordinate system and a set of field-aligned coordinates. Figure 2.1 presents the shape of the magnetic field lines in both geometrical coordinates (the geometrical poloidal angle is defined such that ) and flux coordinates , based on a reconstructed equilibrium of a discharge in the Tore Supra tokamak.

Figure 2.1: Comparison of the shape of magnetic field lines for a single flux-surface in the case of geometrical coordinates (bottom) and flux-coordinates (top). Computation based on the reconstructed equilibrium of the Tore Supra discharge #40816, the average field-line pitch for this flux-surface is .

The variation of with the flux-surface label is called the magnetic shear. It is quantified by the quantity which plays a major role in tokamak physics, for example magnetic configurations with a region of reversed shear (negative ) have been observed to have enhanced confinement properties. Figure 2.2 illustrates this where 3 different surfaces with different values of have been drawn.

Figure 2.2: Different flux surfaces can have different averaged field line pitch.

2.4 The Grad-Shafranov equation

In the absence of toroidal rotation of the plasma or pressure anisotropy, the force balance equation for the equilibrium plasma () is written

(2.17)

where is the current density. Since is constant on flux-surfaces due to the large parallel heat conductivity of electrons,and is therefore only a function of , the and components of this equation imply that the contravariant component of vanishes. For the structure of the magnetic field, this means

(2.18)

which in the axisymmetric case implies that is a function of alone.

The component of equation (2.17) writes

(2.19)

The components of and are expressed as

(2.20)
(2.21)
(2.22)
(2.23)

the last two identities have been obtained using the fact that the toroidal direction is orthogonal to the other two. These are then injected into equation (2.19) to obtain the Grad-Shafranov equation,

(2.24)

Approximate solutions of the Grad-Shafranov equation can be obtained in the case of circular flux-surfaces using an expansion in the ratio of the plasma minor radius and major radius [war66, gre71]. See appendix E for the case of concentric flux-surfaces (first order in ) and section 5.1 for the case of shifted flux-surfaces (second order in ). In an appendix of reference [whi84], the Grad-Shafranov equation is extended to geometries with a helicoidal symmetry.

2.5 Additional definitions

The value of the poloidal flux on the last closed magnetic surface (LCFS) is noted .

The coordinates of the magnetic axis in the system are noted and the amplitude of the magnetic field on the axis is noted . The assumption will be made unless mentioned otherwise. will be referred to as the plasma major radius, and the plasma minor radius is defined as the distance of the magnetic axis to the LCFS on the outboard midplane .

The poloidal angle corresponding to the minimum of the magnetic field amplitude for one given flux-surface is noted and one then defines . Similarly the value of corresponding to the maximum of is noted and one defines .

Two quantities with the same structure as the safety factor are defined:

(2.25)
(2.26)

Chapter 3 Guiding-center motion

Guiding-center theory, introduced by Littlejohn [lit79, lit81] deals with the motion of charged particles in magnetic fields. The description of this motion is simplified by averaging over the fast cyclotronic motion of the particles around the field lines. In the first section of this paragraph the basic assumptions and features of guiding-center motion are recalled. Then the equations of motion for particles evolving in static fields are derived following White et al. [whi03]. Using the results from chapter 2 the case of the tokamak is studied and some basic features of the guiding-center motion applicable to more general magnetic configurations such as the particle drifts are shown. In section 3.4, the motion of charged particles in a tokamak is decomposed in three different motions with well-separated timescales, the expressions of the corresponding frequencies in general flux-surface geometry are recalled as well as their expansion in the case where the Larmor radius of the particles is much smaller than the plasma minor radius (this is called the zero-orbit width limit).

3.1 Charged particle motion lagrangian

The standard form of the lagrangian for the motion of a non-relativistic charged particle of mass and charge , in an electromagnetic field characterized by the potentials and is

(3.1)
(3.2)

from which we can recover the standard result for the canonical momentum .

Littlejohn derived an expression for the guiding-center lagrangian correct to first order in the gyro-radius [lit79, lit81]. This expansion is valid under the assumption that the fields evolve slowly compared to the Larmor frequency and that the characteristic gradient lengths of the fields are greater than the Larmor radius of the particles (where is the magnitude of the velocity component perpendicular to the magnetic field),

(3.3)

where is either one of the electromagnetic field components or of the potentials. We define the parameter which corresponds to the ratio of the Larmor radius and the gradient length of the magnetic field amplitude

(3.4)

then the condition for the validity of the guiding-center theory for equilibrium fields is simply .

The velocity is separated into a parallel and a perpendicular velocity where is the unit vector in the direction of the magnetic field and where are two unit vectors such that and is the gyrophase. The position is written with defined by giving . This expression for uniquely defines the quantity which is called the guiding-center position, the quantity is the gyroradius.

Under the assumptions (3.3) we make the so-called gyrocenter expansion by writing

the expression obtained by Littlejohn for the Lagrangian is

(3.5)
(3.6)

The quantity is the first-order expression of the magnetic momentum which is an adiabatic invariant. The quantity is called the parallel gyroradius. The following expressions of the lagrangian and hamiltonian are correct to first-order in the gyroradius, the associated Euler-Lagrange equations describe the motion of the gyrocenter position.

3.2 Equilibrium motion

3.2.1 Phase-space variables

For the gyrocenter’s position, the same variables are used as for the description of the magnetic field. The phase-space lagrangian , depending on the phase-space variables and their time-derivatives, is expressed using the covariant representations of and , giving

(3.7)

3.2.2 Fast gyromotion

In expression (3.7), all the fields and potentials are not evaluated at the particle’s position but at the gyrocenter’s position. The consequence for the Euler-Lagrange equation is

(3.8)

which is not a surprise since is an adiabatic invariant. If we now consider the dependence over , then one obtains the following equation

(3.9)

which simply tells us that the gyrophase oscillates at the gyrofrequency . In other words, is the angle associated to the gyromotion of the particle and the action which is canonically conjugate to . The term in the lagrangian is further dropped for convenience since it will not affect the equations for the remaining variables.

3.2.3 Equations of motion

The gyrocenter lagrangian can be written

(3.10)

with . The extremalization over the remaining variables yields the following equations

with , , , , and . The equations of motion are then obtained by inverting the matrix.

with . This result is valid for any equilibrium field with nested flux-surfaces such that conditions (3.3) are met.

3.3 The tokamak case

3.3.1 Equations of motion

In the case of an axisymmetric tokamak, is an ignorable coordinate such that all -derivatives vanish. In particular such that Noether’s theorem tells us that is an invariant of motion. Additionally we showed in chapter 2, that , such that .

The equations of motion are

(3.11)
(3.12)
(3.13)
(3.14)

with .

In the following paragraphs, we suppose that the ordering of the static electric field is such that its effect on the motion of the guiding center is only of second order in .

3.3.2 Motion along field lines

Since and , the movement of the guiding center is, to first order in , along the field line,

(3.15)
(3.16)
(3.17)

which is simply .

3.3.3 Magnetic and electric drifts

It is also interesting to look at the second order terms in these equations. One obtains

(3.18)
(3.19)
(3.20)
(3.21)

If one then separates the terms in from the terms in , then one obtains the decomposition , where is the electric drift or drift, is the gradient-B drift and the curvature drift defined by

(3.22)
(3.23)
(3.24)

where is the magnetic field curvature, which expressed in terms of the components of the magnetic field is