Non-linear modelling of saturated internal and external MHD instabilities in tokamaks

Macroscopic instabilities and their unfavourable effects on plasma confinement pose a central challenge for the development of reactor relevant tokamak scenarios. Some promising operation scenarios feature extended regions of low magnetic shear. These are in the core region for hybrid scenarios, and in the pedestal-edge region for e.g. the ELM-free quiescent H-mode (QH-mode). This thesis presents a non-linear study of macroscopic magnetohydrodynamic (MHD) instabilities with a focus on plasmas with regions of low magnetic shear. In the first part, we investigate the triggering of fast growing resistive modes in hybrid tokamak plasmas, where infernal modes can couple to neoclassical tearing modes (NTMs). Numerical simulations with the non-linear resistive initial value code XTOR-2F with and without inclusion of bootstrap current effects allow us to determine the evolution of MHD modes from the linear to the late non-linear phase. An analytical model is developed to describe the vanishing of mode coupling in the early non-linear regime. In this context, we extend the tearing stability parameter $\Delta'$ from the linear to the non-linear phase and calculate the individual contributions to the growth of the resistive mode. This allows for an identification of the triggering mechanism in the initial phase and the dominant terms in the non-linear phase. A comparison with the numerical results shows that infernal mode coupling can destabilise otherwise stable NTMs and thus seed $2/1$ magnetic islands. The helically perturbed bootstrap current is found to further destabilise the magnetic islands in the non-linear phase. In the second part of the thesis, 3D free-boundary equilibrium computations are employed to describe saturated external kink-type modes. The approach is first demonstrated to capture the salient features of non-linearly saturated external kink modes in standard baseline tokamak scenarios and is then applied to QH-mode plasmas. A method to conveniently extract the saturated displacement amplitude from edge-corrugated VMEC equilibria in terms of straight field line Fourier modes is presented. For standard current-driven modes the amplitude is compared with a non-linear analytical model, for which a numerical solver is implemented. In QH-mode plasmas, dangerous edge localised modes (ELMs) are replaced by benign edge harmonic oscillations (EHOs). The latter are commonly assumed to be connected to saturated external kink states. In our study of QH-mode plasmas, we consider two different driving mechanisms for external kink type-modes separately. We find that standard current-driven external kinks are linearly unstable, and non-linearly stable in a wide parameter range, especially where $q_{\text{edge}} \lesssim m/n$ . But, where standard current-driven kinks are linearly stable we find that coupling of pressure-driven infernal modes can cause instability, and their upper sideband drives edge corrugations that appear to have external kink features. Both types of modes are identified with the VMEC equilibrium code, and the spectra are compared favourably with those of linear numerical approaches and analytic methods. EHOs could be connected to both type of modes.

Gyrokinetic particle-in-cell global simulations of ion-temperature-gradient and collisionless-trapped-electron-mode turbulence in tokamaks

Sébastien Jolliet

The goal of thermonuclear fusion research is to provide power plants, that will be able to produce one gigawatt of electricity. Among the different ways to achieve fusion, the tokamak, based on magnetic confinement, is the most promising one. A gas is heated up to hundreds of millions of degrees and becomes a plasma, which is maintained – or confined – in a toroidal vessel by helical magnetic field lines. Then, deuterium and tritium are injected and fuse to create an α particle and an energetic neutron. In order to have a favorable power balance, the power produced by fusion reactions must exceed the power needed to heat the plasma and the power losses. This can be cast in a very simple expression which stipulates that the product of the density, the temperature and the energy confinement time must exceed some given value. Unfortunately, present-days tokamaks are not able to reach this condition, mostly due to plasma turbulence. The latter phenomenon enhances the heat losses and degrades the energy confinement time, which cannot be predicted by analytical theories such as the so-called neoclassical theory in which the heat losses are caused by Coulomb collisions. Therefore, numerical simulations are being developed to model plasma turbulence, mainly caused by the Ion and Electron Temperature-Gradient and the Trapped-Electron-Mode instabilities. The plasma is described by a distribution function which evolves according to the Vlasov equation. The electromagnetic fields created by the particles are self-consistently obtained through Maxwell's equations. The resulting Vlasov-Maxwell system is greatly simplified by using the gyrokinetic theory, which consists, through an appropriate ordering, of eliminating the fast gyromotion (compared to the typical frequency of instabilities). Nevertheless, it is still extremely difficult to solve this system numerically due to the large range of time and spatial scales to be resolved. In this thesis, the Vlasov-Maxwell system is solved in the electrostatic and collisionless limit with the Particle-In-Cell (PIC) ORB5 code in global tokamak geometry. This Monte-Carlo approach suffers from statistical noise which unavoidably degrades the quality of the simulation. Consequently, the first part of this work has been devoted to the optimization of the code with a view to reduce the numerical noise. The code has been rewritten in a new coordinate system which takes advantage of the anisotropy of turbulence, which is mostly aligned with the magnetic field lines. The overall result of the optimization is that for a given accuracy, the CPU time has been decreased by a factor two thousand, the total memory has been decreased by a factor ten and the numerical noise has been reduced by a factor two hundred. In addition, the scaling of the code with respect to plasma size is presently optimal, suggesting that ORB5 could compute heat transport for future fusion devices such as ITER. The second part of this thesis presents the validation of the code with numerical convergence tests, linear (including dispersion relations) and nonlinear benchmarks. Furthermore, the code has been applied to important issues in gyrokinetic theory. It is shown for the first time that a 5D global delta-f PIC code can achieve a thermodynamic steady state on the condition that some dissipation is present. This is a fundamental result as the main criticism against delta-f PIC codes is their inability to deal with long time simulations. Next, the role of the parallel nonlinearity is studied and it is demonstrated in this work that this term has no real influence on turbulence, provided the numerical noise is sufficiently low. This result should put an end to the controversy that recently occurred, in which gyrokinetic simulations using different numerical approaches yielded contradictory results. Finally, thanks to the optimization of the code, the gyrokinetic model has been extended to include the kinetic response of trapped-electrons, in place to the usual adiabatic (Boltzmann) approximation. For the first time, global TEM nonlinear simulations are presented, and the role of the zonal flow on heat transport is analyzed. This study will help in acquiring some knowledge on the less-known TEM turbulence (as compared to ITG). In conclusion, this thesis is one of the main steps of the development of ORB5, which is now a state-of-the-art gyrokinetic code for collisionless ITG and TEM turbulence, and has brought several contributions to the understanding of these phenomena.

EPFL2009

Bootstrap current compensation with electron-cyclotron waves in 3D reactor-size configurations

Sergi Ferrando I Margalet

In magnetic confinement devices, the inhomogeneity of the confining magnetic field along a magnetic field line generates the trapping of particles (with low ratio of parallel to perpendicular velocities) within local magnetic wells. One of the consequences of the trapped particles is the generation of a current, known as the bootstrap current (BC), whose direction depends on the nature of the magnetic trapping. The BC provides an extra contribution to the poloidal component of the confining magnetic field. The variation of the poloidal component produces the alteration of the winding of the magnetic field lines around the flux surfaces quantified by the rotational transform ι. When ι reaches low rational values, it can trigger the generation of ideal MHD instabilities. Therefore, the BC may be responsible for the destabilisation of the configuration. This thesis is divided into two parts. In the first part, we present a self-consistent method to calculate the BC and assess its effect on equilibrium and stability in general 3D configurations. This procedure is applied to two reactor-size prototypes (both with plasma volumes ∼ 1000m3): a quasi-axisymmetric (QAS) system and a quasi helically symmetric (QHS) system with magnetic structures that develop BC in opposite directions. The BC increases with the plasma pressure, therefore its relevance is enhanced when dealing with reactor-level scenarios. The behaviour of both prototypes at reactor level values of β ≡ (kinetic plasma pressure)/(magnetic pressure) is assessed, as well as its alteration of the equilibrium and stability. In the QAS prototype, BC-consistent equilibria have been computed up to β = 6.7% and the configuration is shown to be stable up to β = 6.4%. Convergence of self-consistent BC calculations for the QHS case is achieved only up to β = 3.5%, but the configuration is unstable for β ≥ 0.6%. The relevance of symmetry breaking modes of the Fourier expansion of the confining magnetic field on the generation of BC is studied for each prototype. This proves the close relationship between magnetic structure and BC. Having established the potentially dangerous implication of the BC, principally, in reactor prototypes, a method to compensate its harmful effects is proposed in the second part of the thesis. It consists of the modelling of the current driven by externally launched ECWs within the plasma to compensate the effects of the BC. This method is flexible enough to allow the identification of the appropriate scenarios in which to generate the required CD depending on the nature of the confining magnetic field and the specific plasma parameters of the configuration. Both the BC and the CD calculations are included in a self-consistent scheme which leads to the computation of a stable BC+CD-consistent MHD equilibrium. This procedure is applied in this thesis to simulate the required CD to stabilise the QAS and QHS prototypes introduced in the first part. The estimation of the input power required and the effect of the driven current on the final equilibrium of the system is performed for several relevant scenarios and wave polarisations providing various options of stabilising driven currents. Several scenarios have been devised for each prototype in order to drive current at the appropriate location and with the desired direction. Different polarisations and launching conditions have been employed to this purpose. In particular, a HFS launched X2 ECW with an input power of 1.5MW has been shown to drive sufficient current to maintain the rotational transform below the critical value 2/3 at β = 6.4% for the QAS reactor. Correspondingly, in the QHS reactor, an X3-mode ECW of 100KW was sufficient to drive the current required to push the rotational transform below unity near the magnetic axis at β = 3%. Thus, stabilisation of BC-driven instabilities with externally launched ECWs has been achieved for both contrasting configurations. The method proposed in this thesis allows also the utilisation of EBW in the generation of CD. The possible advantages of EBCD for compensation studies are described as well as their possible application to the two prototypes under consideration. The BC+CD procedure is particularly interesting to investigate new magnetic geometries as potential candidates for fusion reactors. With this numerical tool, it is possible to assess the implications of their consistent BC when operating at reactor level. It also allows to quantify how much power would be required to maintain the system MHD stable in these circumstances. Nevertheless, this method is flexible enough to be applicable to any configuration.

EPFL2006

Stability and toroidal rotation properties of highly shaped plasmas in the TCV tokamak

Andrea Scarabosio

Magnetohydrodynamic (MHD) instabilities and plasma rotation have various impacts on particle and thermal transport in toroidal plasmas. MHD instabilities degrade the confinement, limit the maximum achievable plasma pressure, and can lead to plasma disruptions. Plasma rotation is observed to have beneficial effects on global instabilities and improve confinement through the reduction of turbulent transport. Plasma rotation and stability are strongly coupled, influencing each other in a complicated fashion. The Tokamak à Configuration Variable (TCV), unique in its plasma shaping capabilities, is equipped with a flexible auxiliary heating system (ECRH) and several diagnostics allowing us to study the effect of the plasma shape on stability in a wide range of scenarios. TCV also provides for measurements of carbon toroidal velocity in the absence of external momentum input, an experimental condition poorly studied in the past but of major interest for an accurate prediction of the toroidal rotation in future large experiments. The work presented in this thesis may be divided into two parts. In the first part, we focus on plasma instabilities appearing in three different TCV scenarios. These instabilities limit the achievable maximum pressure and current density. New features and previously unnoticed dependencies are shown. In the second part of this thesis, we experimentally study the plasma rotation properties and their relation with the plasma parameters and MHD activity. The new insight on spontaneous rotation may help in constructing a complete model of plasma rotation in tokamaks. During the initial plasma current rise, edge MHD instabilities are commonly observed in most tokamaks when the edge safety factor qa approaches low rational values. These instabilities may lead to plasma disruptions and can effectively limit the edge safety factor to qa > 3, which is above the well-known operational limit qa ≥ 2. We report on a detailed analysis of the MHD modes leading to disruptions and show experimental evidence of the key role of mode coupling. The beneficial effect of plasma shaping on current rise experiments was early noted in the TCV operation. In this thesis we characterise the effect of the plasma shape on the instability, showing how plasma elongation and triangularity (positive and negative) act as stabilising factors. The complex shape dependence of such MHD modes is interpreted on the basis of the theory of coupled tearing modes. Three stabilising mechanisms linked with plasma shaping are shown to be important for TCV and likely to other tokamaks. Sawteeth, which are central relaxation oscillations common to most tokamak scenarios, also have a significant effect on central plasma parameters. In highly elongated TCV discharges heated with far off-axis ECRH, the sawtooth oscillations are observed to disappear and to be replaced by a continuous MHD mode resonant on the q = 1 surface. The combination of a flat current profile and small q = 1 radius determines the change in the plasma stability, which we study using ideal and resistive MHD models. Plasmas with internal transport barriers (ITBs) are routinely produced in TCV using high power ECRH and current drive. A variety of MHD activity is observed during these discharges including classical m/n = 3/1 and neoclassical m/n = 2/1 tearing modes, and more exotic q = 2 pseudo-sawteeth central relaxations. Disruptive modes are observed in ITB plasmas with strong reverse magnetic shear. In general, the MHD instabilities limit the maximum achievable plasma pressure. The experimental β-limit in this scenario is observed to depend strongly on the peaking of the electron pressure profile, in agreement with the ideal MHD theory. The properties of the toroidal carbon impurity rotation in Ohmic limited L-mode plasmas are studied in detail in stationary conditions. The dependence and scaling of the toroidal velocity with plasma parameters, such as the plasma current and density, are highlighted, as well as the effect of the sawtooth activity on the rotation profile. We show that the toroidal rotation in TCV is generally directed in the counter-current or electron diamagnetic drift direction. We also compare the experimental results with the neoclassical prediction for stationary toroidal rotation in absence of external momentum input. Angular momentum relaxations are observed in TCV Ohmic plasmas. Large magnetic islands cause strong losses of angular momentum, flattening the rotation profile. Once the MHD mode has disappeared, the stationary angular velocity profile is restored over a typical time scale, 100–200 ms, providing experimental evidence of the spontaneous torque spinning the plasma column. We show that we can reproduce the phenomena of "toroidal spin-up" and rotation inversion observed in TCV Ohmic plasmas with a phenomenological momentum transport model. We infer transport coefficients, such as the momentum diffusivity, and compare them with theoretical predictions.

EPFL2006

Bootstrap current compensation with electron-cyclotron waves in 3D reactor-size configurations

Sergi Ferrando I Margalet

EPFL2006

Stability and toroidal rotation properties of highly shaped plasmas in the TCV tokamak

Andrea Scarabosio

EPFL2006

Gyrokinetic particle-in-cell global simulations of ion-temperature-gradient and collisionless-trapped-electron-mode turbulence in tokamaks

Sébastien Jolliet

EPFL2009