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Publication# Computation of linear MHD instabilities with Multi-region Relaxed MHD energy principle

Résumé

Over the last decade, a variational principle based on a generalisation of Taylor's relaxation, referred to as multi-region relaxed magnetohydrodynamics (MRxMHDs) has been developed. The numerical solutions of the MRxMHD equilibria have been constructed using the Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502). In principle, SPEC could also be established to describe the MRxMHD stability of an equilibrium. In this work, a theoretical framework is developed to relate the second variation of the energy functional to the so-called Hessian matrix, enabling the prediction of MHD linear instabilities of cylindrical plasmas, and is implemented in SPEC. The negative and positive eigenvalues of the Hessian matrix predict the stability of an equilibrium. Verification studies of SPEC stability results with the M3D-C1 code and the tearing mode $\Delta^{^{\prime}}$ criterion have been conducted for ideal and resistive MHD instabilities, respectively, in a pressureless cylindrical tokamak, and have shown good agreement. Our stability analysis is a critical step towards understanding the MHD stability of three-dimensional MHDs where nested flux surfaces, magnetic islands and stochastic regions co-exist.

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Flux (physique)

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Plasma stability

The stability of a plasma is an important consideration in the study of plasma physics. When a system containing a plasma is at equilibrium, it is possible for certain parts of the plasma to be dist

Neoclassical tearing modes (NTMs), magnetic islands located at rational $q$ surfaces, are an important class of resistive magnetohydrodynamics (MHD) instabilities in tokamak plasmas, with $q$ the safety factor. NTMs are one of the main constraints of the achievable plasma pressure by increasing the local radial transport and NTMs can lead to plasma disruptions. It is therefore crucial to understand the physics of NTMs and ensure their reliable control.
This thesis explores the physics and control of NTMs, by means of dedicated experiments in the TCV tokamak and interpretative simulations with the modified Rutherford equation (MRE), a model widely used in interpreting island width evolutions. Triggerless NTMs originating from unstable tearing modes (TMs, stability index $\Delta'>0$) and saturating under the effects of the perturbed bootstrap current are the main focus of this thesis.
In TCV, triggerless NTMs are reproducibly observed in low-density discharges with strong near-axis electron cyclotron current drive (ECCD), providing an excellent opportunity of studying these modes. Instead of direct computations of $\Delta'$, a model for $\Delta'$ is developed based on extensive experiments and interpretative simulations. This model facilitates the clarification of the complete evolution of triggerless NTMs, from onset as TMs to saturation as NTMs. Our $\Delta'$ model also explains an unexpected density dependence of the onset of NTMs, where NTMs only occur with a certain range of density that broadens with increasing near-axis ECCD power and with lower plasma current. The density range is found to result from the density and plasma current dependence of the stability of ohmic plasmas and the density dependence of ECCD efficiency.
Given its high localization and flexibility, off-axis ECH/ECCD will be used for NTM control in future tokamaks. Comprehensive experimental and numerical studies of the dynamics of NTMs are carried out in this thesis, concerning both the stabilization of existing NTMs and the prevention of NTMs by means of preemptive off-axis ECCD. It is shown and predicted that the prevention of NTMs is much more efficient than NTM stabilization in terms of EC power. Interpretative simulations of the complex set of experiments constrain well the coefficients in the MRE and quantify NTM evolutions. The prevention effects from off-axis ECCD are found to result from local ECH/ECCD instead of a change of $\Delta'$.
A key element of a reliable real-time (RT) control of NTMs is the alignment of EC beams with the target mode location. A small sinusoidal sweeping of the deposition location of EC beams around the target location proves to be effective for both NTM stabilization and prevention, making it a promising technique. Integrated control of NTMs, plasma pressure and model-estimated $q$ profiles is demonstrated on TCV, including advanced plasma state reconstruction, monitoring, supervision and actuator management. A RT-capable MRE module, based on our validated MRE, is developed for the first time and tested by extensive offline simulations for TCV and AUG. It provides a more intelligent physics-based NTM controller, aware of the EC power it requires to stabilize or prevent a given NTM. The information from the RT-MRE is also valuable for RT actuator allocations and decision-making in view of the overall integrated control, in particular for future long-pulse tokamaks like ITER and DEMO.

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.

High-beta, low-aspect-ratio ("compact") stellarators are promising solutions to the problem of developing a magnetic plasma configuration for magnetic fusion power plants that can be sustained in steady state without disrupting. These concepts combine features of stellarators and advanced tokamaks and have aspect ratios similar to those of tokamaks (2-4). They are based on computed plasma configurations that are shaped in three dimensions to provide desired stability and transport properties. Experiments are planned as part of a program to develop this concept. A beta = 4% quasi-axisymmetric plasma configuration has been evaluated for the National Compact Stellarator Experiment (NCSX). It has a substantial bootstrap current and is shaped to stabilize ballooning, external kink, vertical, and neoclassical tearing modes without feedback or close-fitting conductors. Quasi-omnigeneous plasma configurations stable to ballooning modes at beta = 4% have been evaluated for the Quasi-Omnigeneous Stellarator (QOS) experiment. These equilibria have relatively low bootstrap currents and are insensitive to changes in beta. Coil configurations have been calculated that reconstruct these plasma configurations, preserving their important physics properties. Theory- and experiment-based confinement analyses are used to evaluate the technical capabilities needed to reach target plasma conditions. The physics basis for these complementary experiments is described. (C) 2000 American Institute of Physics. [S1070-664X(00)94905-X].

2000