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Basic introduction to tokamak plasma description
In this section, the physical fundamentals for the description of the magnetized plasma behaviour are given. Much more comprehensive details on the following models and equations can be found in well-known manuals about magnetized plasma physics and tokamak devices, such as [8, 9, 10, 11] and references therein.
The magnetic equilibrium in tokamaks
It is now briefly introduced the magnetic field equilibrium in a tokamak device and the coordinates that define it. Being the magnetic field in a tokamak a con-tinuous vector field, it is possible to determine the set of tangent vectors, which are usually called magnetic field lines. In order for the charged particles in a plasma to be confined by the field lines, those must lie on a closed surface, which are termed as magnetic or flux surfaces. In a tokamak, a magnetic surface is topologically a torus, and therefore a convenient set of coordinates to describe this geometry is the toroidal coordinates x = ( ; ; ’), where is the label of the magnetic surface, whereas and ’ are respectively the poloidal and toroidal (ax-isymmetric) angles. Actually, as it is described later, the magnetic field lines must have both a toroidal and a poloidal components in order for the charged particles to be confined. A schematic view of a circular cross-section toroidal geometry is illustrated in Figure 1.2. Being the tokamak configuration axisymmetric around the vertical central axis, the equilibrium magnetic field can be written as: Beq = r r’ + I( )r’ (1.4).
Gyrokinetic theory
As already stated, the statistical kinetic description of the particle trajectories in a magnetized plasma is extremely expensive for the present computational resources. Nevertheless, the resonant mechanisms taking place in the veloc-ity space must be taken into account in order to retain the relevant physics for studying the tokamak plasma core, and in particular the results reported in this thesis. For this reason, a well-established theoretical framework is now briefly introduced, which allows to reduce the 6-D system of equations of the kinetic ap-proach to five dimensions. Such a framework is the gyrokinetic approach [23], which will be described in the following. Yet, there could be other interesting solutions to the problem, which rely on the so-called hybrid approach. These are based on the fluid description of the thermal background plasma, while describ-ing a selected species with a fully kinetic model. An example of a numerical code adopting this modeling framework is JOREK [24, 25, 26].
The gyrokinetic theory is based on the adiabatic limit. To fully comprehend the adiabatic limit, it is important firstly to briefly introduce the motion of a charged particle embedded in a magnetic field. Due to the Lorentz force 1.17, the charged particles are constrained to follow the magnetic field lines describing helical trajectories around it. Projecting this trajectory on a plane perpendicular to the magnetic field line, the motion can be seen as a superposition of a guiding-center motion and a gyromotion. Basically, while it is moving along the field line, the particle is rotating around the field line at the so-called cyclotron fre-quency !c = jqB=mj with a circular trajectory whose radius is the Larmor radius L = mv?=(qB). The motion variables can then be written as:
x(t) = X(t) + L(X; v?; ’c; t) (1.37).
v(t) = vkb(X; t) + v?(X; ’c; t).
with the term X representing the guiding center position in space and ’c the gy-rophase. It is relevant now to determine that the characteristic time of the gyro-motion and the spatial extension of the ion Larmor radius are much smaller than the characteristic time and spatial scales of the electromagnetic field variations in the whole plasma domain. These conditions can be summarized as: !EF @t @t !c @ log B @ log E.
Orbits of charged particles in toroidal plasmas
In this section, the orbits of charged particles in magnetized toroidal plasmas will be addressed within the already introduced adiabatic limit theory. The par-ticle orbits are considered in the guiding-center coordinates. In the axisymmetric toroidal configuration and neglecting the perturbations of the electromagnetic fields, which excludes the turbulent regimes, it is possible to show that the parti-cle motion can be described by three invariants along the trajectory of the parti-cles. The three invariants of the motion of a charged particle plunged in a static electromagnetic field, within the adiabatic approximation, are the total energy E, the magnetic moment and the toroidal kinetic momentum P’.
As it is explained in section 1.2.1, the strength of the magnetic field varies with the major radius R of the tokamak as B 1=R. Moreover, the magnetic field line lies on the magnetic field surface, and therefore explores the whole poloidal angle in turning around the torus. The same does the charged particle confined by this magnetic field, which hence experiences the variation of the magnetic field in its trajectory, namely the peak and the wells of the magnetic field. The total energy of the particle, in the absence of electrostatic potential is E = msv2=2. The velocity in this latter relation can be decomposed in parallel and perpendicular directions and rewritten in terms of the magnetic moment as E = msvk2=2 + B. Making explicit the parallel velocity: vk = r (1.44) ms (E B).
It should be reminded once again that the electrostatic potential has been con-sidered = 0. It is therefore straightforward that, depending on the value of its energy E, the particle can be trapped in specific regions where the term B describes a well of potential. In other words, when the relation » v? # max (1.45) E vk 2 =B 1+ B.
Microinstabilities driven by the thermal particles
The microinstabilities driven by the thermal particles can be further classified based on the excitation mechanism. After describing the fundamental mecha-nism of the drift wave, a heuristic explanation of the drift-wave and interchange instabilities will be given in the following paragraphs. Both instabilities are gen-erally present in tokamak plasmas, and while the latter occurs in the 2-D flute approximation (i.e. along the magnetic field line direction), the former instability requires a three dimensional description. Then, the principal microinstabilities responsible of the anomalous transport in tokamaks, and also important for the comprehension of the remainder of the thesis, are described and treated individ-ually.
The drift wave is a perturbation originated by the fluctuations of the electrostatic potential and the particle density, which can concern both electrons and ions. In this context, the adiabatic electron response is a relevant approximation. It as-sumes that the electrons, which have a very low inertia, promptly accommodate any perturbation of the electrostatic potential, which can be expressed by a plane wave of the form ei(kyy !t). Thus, in the limit ! kkvth;e, the electron density perturbation ne is in phase with the electrostatic potential perturbation , and the fast motion of the electrons along the parallel direction allows the Boltzmann description of the electron response to the electrostatic potential perturbation: ne ’ e (1.49) ne Te.
Brief history on the impact of fast ions on micro-turbulence
As a final additional section, a brief background on earlier studies performed on the impact of fast ions on microturbulence in tokamak plasmas will be given. The main goal of this latter section is to plunge the reader into the right and exhaustive context in order to evaluate the results that will be mainly presented in Chapters 3 and 4.
Although the impact of fast ions, and more in particular of fusion-born al-pha particles, on the plasma confinement is a long-standing issue [65], some first clear experimental observations of the possible interaction between microturbu-lent transport and externally generated fast ions were produced in both JET L-mode [66, 67] and H-mode [68] confinement plasmas. Firstly hypothesized to be related to the perpendicular shearing generated by the substantial amount of tan- gential NBI power introduced in the plasmas or dilution effects [69], the decrease of the thermal ion stiffness was eventually strongly linked to the poloidal plasma current and the p [70]. This latter parameter is defined as the ratio between the thermodynamical pressure over the magnetic pressure generated by the poloidal magnetic field. Indeed, for the sake of clarity, the stiffness of the temperature pro-file is basically the relation between the local ion heat flux with the normalized ion temperature gradient, intimately related to the turbulent transport levels.
The observation of the strong effect of p on the confinement was followed by the intuition on the crucial role of the externally generated fast ions, whose high energy strongly contributes to the value of p. In this context, the gyroki-netic simulations started to be the fundamental tool to deeply investigate such a physical mechanism leading to the ion stiffness decrease. In fact, numerical analyses with the gyrokinetic code GENE [57] highlight the strong impact of the suprathermal species on the ion turbulent flux, which was reduced in the non-linear regime down to experimental values only in simulations retaining the fast ions and the fluctuations of the magnetic potential [71]. Already in those studies, the beneficial effects of the fast ions on the linear growth rate, although of minor importance with respect to the nonlinear effects, were also observed. Such linear effects were firstly traced back to the dilution and to the increase of the Shafra-nov shift, both having well-known positive impact on the linear ITG excitation [12], but the growth rate reduction, and the subsequent heat flux reduction, over-estimated the theoretical prediction of those two latter effects. The linear wave-wave interaction between the ITG mode frequency and the magnetic drift of the suprathermal particles was established to be the main cause of this growth rate reduction [72]. Such a linear electrostatic mechanism was achieved and further optimized in ICRH-heated plasmas for a narrow range of fast ion temperature, beneficially impacting at the local ratio Tfast=Te 10 and strongly decreasing its significance at higher fast ion temperature [73]. Although not effective thereby for fusion-born alpha particles in burning plasmas, this beneficial impact was ex-perimentally validated firstly in L-mode plasmas with strong ICRH power at JET [74], and more recently found to be strongly active in building Internal Transport Barriers in ASDEX-U scenarios [75], leading thus to a turbulent transport reduc-tion.
A crucial point throughout the gyrokinetic studies on the fast-ion impact on ITG transport was the destabilization of large-scale high-frequency modes ex-cited by the suprathermal particles. Such modes are mainly electromagnetic in-stabilities, driven by the fast-ion pressure gradient, by the parameter or by a combination of those two parameters, nonetheless they were never detected in the experiments. At this point, it should be stressed the fact that fast-ion charac-terization is not an easy task, and poorly accurate modeling can only be obtained. Therefore, any inference on the local gradients or fast-ion parameters reveals to be affected by large error bars. Furthermore, in most of the integrated modeling suites, the module dedicated to the computation of the fast ion distribution func-tion does not take into account the anomalous transport of suprathermal particles induced by fast-ion-driven MHD modes, and therefore an overestimation of the fast ion confinement could be also expected. For this reason, the fast ion pres-sure gradients employed in the subsequent gyrokinetic studies must be further assessed and validated by large scans, which could be, sometimes, not afford-able. For this reason, the issue related to the destabilization of fast-ion-driven modes has been overcome in the modeling by lowering the pressure gradient [76, 77], i.e. the drive of the high-frequency instabilities. Beyond the experimen-tal absence of the modes, such an artificial stabilization was also required for matching the power balances computed by the integrated modeling. In fact, in the parameter setup with unstable fast-ion modes, the resulting fluxes from the gyrokinetic simulations were largely beyond those computed via integrated mod-eling [76, 78]. Especially, the electron confinement was strongly degraded, with the magnetic transport extremely enhanced by the large-amplitude fluctuations of the electromagnetic fields induced by the destabilized fast-ion modes. Such a degradation of the electron confinement has also been consistently observed in spherical tokamak experiments with fully destabilized AEs [79].
Table of contents :
1 Introduction to plasma physics related to microinstabilities, fast-iondriven modes and self-regulation mechanisms
1.1 Nuclear fusion
1.1.1 Magnetically confined plasmas
1.2 Basic introduction to tokamak plasma description
1.2.1 The magnetic equilibrium in tokamaks
1.2.2 Modeling of magnetized plasmas towards the f approach .
1.2.3 Fluid moments
1.2.4 Wave-particle interaction
1.2.5 Gyrokinetic theory
1.2.6 Orbits of charged particles in toroidal plasmas
1.3 Introduction to instabilities in tokamaks
1.3.1 Microinstabilities driven by the thermal particles
1.3.2 Instabilities driven by the fast ions
1.4 Nonlinear regime of plasma dynamics
1.4.1 Introduction to turbulence
1.4.2 Nonlinear saturation mechanisms
1.5 Brief history on the impact of fast ions on microturbulence
1.6 Thesis outline
2 Flux-tube simulations of tokamak plasmas
2.1 Flux-tube approach
2.2 The GENE code
2.2.1 System of equations solved in GENE
2.2.2 Diagnosed quantities
3 Impact of MeV ions on ITG-driven turbulence
3.1 Experimental observations in JET three-ion heating scheme scenario
3.1.1 Three-ion heating scheme: Generation of MeV-range ion population
3.1.2 Impact of MeV-range ions on global plasma confinement
3.1.3 Three-ion scheme as effective actuator of Alfv´en activity .
3.2 Numerical setup
3.3 Linear stability studies
3.3.1 Linear stability of the system without fast ions
3.3.2 Impact of fast ions on the linear stability of JET pulse #94701 77
3.3.3 Identification of the fast-ion-driven instability
3.3.4 Study of the TAE excitation mechanism
3.4 Ion-scale turbulence suppression via complex mechanism
3.4.1 Assessment of the nonlinear impact of fast ions on the thermal confinement in JET #94701
3.4.2 Strong TAE-induced fast-ion transport
3.4.3 ITG turbulence suppressed by enhanced zonal activity
3.4.4 Nonlinear coupling between TAE and zonal flow spatiotemporal scales
3.4.5 Low-frequency modes excited by fully destabilized TAEs
3.4.6 Study of the MeV-ion effect on the cross-phase
3.5 Residual electron transport in the presence of zonal fields
3.5.1 Strong zonal field activity in the presence of fully destabilized AEs
3.5.2 Destabilization of high-frequency electron-driven modes
3.6 Partial conclusions
3.7 Further analyses: Negative magnetic shear effect on fast ion confinement
4 Impact of fast ions on different turbulent regimes: TEM
4.1 JT-60U hybrid scenario
4.2 Numerical setup
4.3 Linear stability studies
4.3.1 Analysis of the linear spectrum without fast ions
4.3.2 Analysis of the linear spectrum with fast ions
4.3.3 Validity of the flux-tube approximation
4.3.4 Linear effects of fast ions on TEM
4.4 Preliminary nonlinear analyses: Need for setup modification
4.4.1 Effects of high-frequency instability on the heat transport
4.5 Fast ion impact on TEM-dominated transport
4.5.1 Description of the simulation setup modifications
4.5.2 Effects of the modified setup on the linear stability
4.5.3 Impact of fast ions on TEM-induced fluxes in the nonlinear regime
4.5.4 Role of zonal flows as saturation mechanism of TEM turbulence
4.6 Partial conclusions
4.7 Numerical experiment: TEM transport reduced with highly energeticions and low- conditions
5 Conclusions and future directions
5.1 Main conclusions
5.2 Future work
Bibliography
