3D ideal linear peeling ballooning theory in magnetic fusion devices

Nuclear fusion is the fundamental process that generates heat and light in the stars but it is also a promising potential candidate for the generation of energy by man. However, where in the center of stars the combination of extreme temperatures with extreme pressure is what drives light elements close enough together for them to fuse and release part of their combined mass as energy, on earth only extreme temperatures can be employed. Matter at these temperatures exists in the state of plasma, where the atoms are stripped clean of their electrons. In the resulting physical system the presence of long term electromechanical forces between the charged particles can lead to violent collective behavior. Therefore, the general question of confining hot plasma in a stable way is crucial in order to achieve fusion. One strategy of doing this is by employing powerful magnetic fields to guide the charged particles around a toroidal configuration. This work is about a class of instabilities that these configurations are susceptible to, called high-n instabilities. High-n instabilities are instabilities that have strong localization around the magnetic field lines that confine the plasma, and they have previously been identified as possible culprits for some significant processes that occur in magnetic configurations, such as the periodic release of energy through Edge-Localized Modes (ELMs), or even the complete loss of confinement during disruptions, during which a large amount of energy is released to the reactor walls, damaging them. However, whereas much work has been performed in this field, the analysis of high-n instabilities in realistic 3-D geometries, including the effects of the deformation of the plasma edge, has not yet been done yet in a systematic and dedicated manner. Therefore, in the first part of this work a suitable theoretical framework is developed. Here, the simplification can be made that only modes pertaining to the same field line couple, through their high-n nature. This reduces the dimensionality of the problem by one, but at the same time does not pose any limitations on the 3-D aspects of the instabilities. One of the results of the theory is a system of coupled ordinary differential equations that can be solved for an eigenvalue, the sign of which determines whether the mode formed by the corresponding eigenvector is unstable or not. The solution of these equations, however, is something that has to be done using numerical techniques, so to this end the numerical code PB3D is developed. This stands for Peeling-Ballooning in 3-D, two modes that are described well through high-n theory. PB3D can treat the stability of various equilibrium codes such a VMEC and HELENA in a modular way, is parallelized making use of the message-passing interface (MPI) and is optimized for speed. The code is verified making use of physical criteria and by comparisons with two other, well-established numerical codes that have ranges of applicability bordering on that of PB3D. The first one, MISHKA, is a general-n stability code for axisymmetric equilibria, whereas the second one, COBRA, can treat general 3-D cases, but only in the n→ ∞ limit, with a static edge. The successful introduction of PB3D paves the way for a multitude of potential applications concerning 3-D edge effects. It can be investigated, for example, how many previous findings concerning peeling-ballooning modes in axisymmetric configurations change or not when 3-D effects are introduced. The theory of high-n stability of axisymmetric equilibria, for example, in the past has shed light on the dynamics of ELMs, and how this changes by including 3-D effects is a topic of interest. This is true even more so as recently the relevance of ELM control has risen due to the potentially dangerous behavior of ELMs in the next generation nuclear fusion reactors. A strategy for controlling them also intrinsically relies on applying 3-D resonant magnetic perturbations. The study of these effects with PB3D is planned in the near future in the ITER Organization. Before that, in this work, as a first concrete application, the modification of the stability of the pedestal of a High-confinement plasma equilibrium configuration by a toroidal field ripple is considered. These so-called H-mode configurations are characterized by a steep pressure gradient near the plasma edge, called the pedestal, which increases the temperature and pressure attainable in the core. Therefore, they are often seen as vital in order to achieve fusion. In practice, however, a degradation of the pedestal size is often observed, due to 3-D modifications of the equilibrium, such as the periodic ripple in the toroidal magnetic field due to the discreteness of the toroidal field coils. It was observed here that the application of a toroidal ripple in the shape of the poloidal cross section in the order of a percent, lead to a substantial decrease in the highest possible pedestal pressure, in the order of 30-40%. This substantiates good qualitative agreement with experimental results, where degradations of similar magnitude were observed.This research was sponsored in part by DGICYT (Dirección General de Investigaciones Científicas y Tecnológicas) of Spain under Project No. ENE2012-38620-C02-02 and Project. No. ENE2015-68265, and also in part by EUROFUSION-WP14-EDU and through FUSENET mobility funding.Programa Oficial de Doctorado en Plasmas y Fusión NuclearPresidente: Nicolas Joost Lopes Cardozo.- Secretario: Eduardo Antonio Ahedo Galilea.- Secretario: Carlos Hidalgo Ver

PB3D: A new code for edge 3-D ideal linear peeling-ballooning stability

A new numerical code PB3D (Peeling-Ballooning in 3-D) is presented. It implements and solves the intermediate-to-high-n ideal linear magnetohydrodynamic stability theory extended to full edge 3-D magnetic toroidal configurations in previous work [1]. The features that make PB3D unique are the assumptions on the perturbation structure through intermediate-to-high mode numbers n in general 3-D configurations, while allowing for displacement of the plasma edge. This makes PB3D capable of very e cient calculations of the full 3-D stability for the output of multiple equilibrium codes. As first verification, it is checked that results from the stability code MISHKA [2], which considers axisymmetric equilibrium configurations, are accurately reproduced, and these are then successfully extended to 3-D configurations, through comparison with COBRA [3], as well as using checks on physical consistency. The non-intuitive 3-D results presented serve as a tentative first proof of the capabilities of the code.This research was sponsored in part by DGICYT (Dirección General de Investigaciones Científicas y Tecnológicas) of Spain under Project No. ENE2015-6826

Three-dimensional linear peeling-ballooning theory in magnetic fusion devices

Ideal magnetohydrodynamics theory is extended to fully 3D magnetic configurations to investigate the linear stability of intermediate to high n peeling-ballooning modes, with n the toroidal mode number. These are thought to be important for the behavior of edge localized modes and for the limit of the size of the pedestal that governs the high confinement H-mode. The end point of the derivation is a set of coupled second order ordinary differential equations with appropriate boundary conditions that minimize the perturbed energy and that can be solved to find the growth rate of the perturbations. This theory allows of the evaluation of 3D effects on edge plasma stability in tokamaks such as those associated with the toroidal ripple due to the finite number of toroidal field coils, the application of external 3D fields for elm control, local modification of the magnetic field in the vicinity of ferromagnetic components such as the test blanket modules in ITER, etc.This research was sponsored in part by DGICYT (Dirección General de Investigaciones Científicas y Tecnológicas) of Spain under Project No. ENE2012-38620-C02-02 and also in part by Comunidad de Madrid Project No. S2009/ENE-1679.Publicad

Influence of toroidal ripple on peeling-ballooning stability in the pedestal of H-mode plasmas

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Progress in the modelling of 3-D effects on MHD stability with the PB3D numerical code and implications for ITER

Introduction\u3cbr/\u3eThe theory of magnetohydrodynamics (MHD) is valuable because it leads to baseline considerations for toroidal magnetic configurations, even when the parameter ranges in which these configurations are situated often don’t strictly satisfy the assumptions behind the MHD theory. The reason for this lies in the strong anisotropy of these configurations, where the dynamics perpendicular to the magnetic field lines is often indeed well-described by the theory, even though the direction parallel to the magnetic\u3cbr/\u3efield lines is not. This work is situated in the study of fluted or high-n modes, which are normal modes that fit in the theory of MHD stability, that show very fast variation accross the magnetic field lines, as compared to the behavior along them. High-n MHD stability is important by itself as it can describe phenomena that are known to be important for the current and next generation of nuclear fusion devices, such as ELMs, which can be interpreted as due to two types of high-n instabilities: ballooning modes and peeling modes. In the view of studying these phenomena in enough detail, the two ingredients that this work combines within the world of high-n stability are the inclusion of a possible vacuum perturbation,\u3cbr/\u3ewhich is necessary for peeling modes to exist in the absence of resistivity; and the correct treatment of 3-D effects, which are important not only for stellarators, but also for tokamaks. An example thereof can be found in the well-known consequences that toroidal field (TF) ripples can have on confinement [Sai+07; Wey+17]; but also in the application of resonant magnetic perturbation (RMP) coils to control ELMs by destabilizing them [Eva+06]. After briefly summarizing the theory of ideal linear 3-D MHD\u3cbr/\u3estability and the advancements of the PB3D (Peeling-Ballooning in 3-D) code over the past year in 2, this work then treats its application to the study of 3-D ballooning stability when applying RMPs in tokamaks\u3cbr/\u3ein 3 and 4. Finally, in section 5, conclusions are phrased as well as the plans for future work

PB3D: a new code for edge 3-D ideal linear peeling-ballooning stability

A new numerical code PB3D (Peeling-Ballooning in 3-D) is presented. It implements and solves the intermediate-to-high-n ideal linear magnetohydrodynamic stability theory extended to full edge 3-D magnetic toroidal configurations in previous work [1]. The features that make PB3D unique are the assumptions on the perturbation structure through intermediate-to-high mode numbers n in general 3-D configurations, while allowing for displacement of the plasma edge. This makes PB3D capable of very efficient calculations of the full 3-D stability for the output of multiple equilibrium codes. As first verification, it is checked that results from the stability code MISHKA [2], which considers axisymmetric equilibrium configurations, are accurately reproduced, and these are then successfully extended to 3-D configurations, through comparison with COBRA [3], as well as using checks on physical consistency. The non-intuitive 3-D results presented serve as a tentative first proof of the capabilities of the code

Evaluation of core beta effects on pedestal MHD stability in ITER and consequences for energy confinement

\u3cp\u3eThe maximum stable pedestal pressure has been shown to increase with core pressure and in combination with profile stiffness this can lead to a positive feedback mechanism. However, the effect is shown to saturate for high β in ASDEX-Upgrade [1]. This paper investigates whether this effect appears in ITER scenarios, using ideal MHD numerical codes HELENA and MISHKA for different ITER scenarios from inductive 7.5-15 MA plasmas to steady-state scenarios at 10 MA. No pedestal pressure saturation is found for inductive scenarios; on the contrary for the 10MA steady-state scenario the pedestal pressure is the same for a wide range of total β and is limited by low n kink-peeling modes. Finally, a comparison of the achievable pressure for various levels of core profile stiffness is made with the IPB98(y,2) scaling law.\u3c/p\u3

Advances in modelling of plasma pedestal behaviour and ELM control in ITER reference plasma scenarios,

Progress in the modelling of the edge pedestal plasma behaviour in ITER plasmas including linear and non-linear MHD stability analysis, ELM triggering by pellet injection and vertical plasma oscillations and ELM control by the application of 3-D fields is described. These activities are implemented under the framework of ITER Scientist Fellow Network Pedestal Group to improve the understanding of the physics processes that dominate ITER pedestal plasmas thus providing a firmer physics base to evaluate the edge plasma properties in ITER H-mode plasmas and for the physics-based\u3cbr/\u3eextrapolation of results obtained in present experiments to ITER