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

Physics-regularized neural network of the ideal-MHD solution operator in Wendelstein 7-X configurations

The stellarator is a promising concept to produce energy from nuclear fusion by magnetically confining a high-pressure plasma. In a stellarator, the confining field is three-dimensional, and the computational cost of solving the 3D MHD equations currently limits stellarator research and design. Although data-driven approaches have been proposed to provide fast 3D MHD equilibria, the accuracy with which equilibrium properties are reconstructed is unknown. In this work, we describe an artificial neural network (NN) that quickly approximates the ideal-MHD solution operator in Wendelstein 7-X (W7-X) configurations. This model fulfils equilibrium symmetries by construction. The MHD force residual regularizes the solution of the NN to satisfy the ideal-MHD equations. The model predicts the equilibrium solution with high accuracy, and it faithfully reconstructs global equilibrium quantities and proxy functions used in stellarator optimization. The regularization term enforces that the NN reduces the ideal-MHD force residual, and solutions that are better than ground truth equilibria can be obtained at inference time. We also optimize W7-X magnetic configurations, where desiderable configurations can be found in terms of fast particle confinement. This work demonstrates with which accuracy NN models can approximate the 3D ideal-MHD solution operator and reconstruct equilibrium properties of interest, and it suggests how they might be used to optimize stellarator magnetic configurations.Comment: 46 pages, 23 figures, to be submitted to Nuclear Fusio

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

Prediction of Fishbone Linear Instability in Tokamaks with Machine Learning Methods

A machine learning based surrogate model for fishbone linear instability in tokamaks is constructed. Hybrid simulations with the kinetic-magnetohydrodynamic (MHD) code M3D-K is used to generate the database of fishbone linear instability, through scanning the four key parameters which are thought to determine the fishbone physics. The four key parameters include (1) central total beta of both thermal plasma and fast ions, (2) the fast ion pressure fraction, (3) central value of safety factor $q$ and (4) the radius of $q=1$ surface. Four machine learning methods including linear regression, support vector machines (SVM) with linear kernel, SVM with nonlinear kernel and multi-layer perceptron are used to predict the fishbone instability, growth rate and real frequency, mode structure respectively. Among the four methods, SVM with nonlinear kernel performs very well to predict the linear instability with accuracy $\approx$95%, growth rate and real frequency with $R^2\approx$98%, mode structure with $R^2\approx$98%.Comment: 28 pages,19 figure

Computation of multi-region relaxed magnetohydrodynamic equilibria

We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented.The authors gratefully acknowledge support of the U.S. Department of Energy and the Australian Research Council, through Grants DP0452728, FT0991899, and DP110102881

Beyond Axisymmetry in Tokamak Plasmas

H-mode tokamak plasmas are characterised by quasi-periodic instabilities, called edge localised modes (ELMs), driven by unstable peeling-ballooning modes inside the pedestal region. For large scale tokamaks, like ITER, the resulting particle and heat fluxes are predicted to be unacceptable and ELM control methods are required. One promising method relies on the application of 3D resonant magnetic perturbations (MPs), and ELM mitigation or even complete suppression is observed. A computational framework is presented that aims to understand the effect of MPs on both plasma equilibria and stability. The ELITE stability code is used to find the linearised plasma response, i.e. the 3D part of the equilibrium, and compute the axisymmetric peeling-ballooning eigenmodes. This information is used to calculate the 3D stability under a perturbative and a variational formulation of the MHD energy principle. In practice, the axisymmetric peeling-ballooning modes are used as trial functions for the minimisation of the 3D energy functional. The symmetry breaking of the toroidal geometry leads to the coupling of toroidal modes which has a direct impact on the linear growth rates of unstable peeling-ballooning modes. This mechanism results in the modification of the plasma stability above a critical value of the applied MP field and field-line localisation of the peeling-ballooning eigenmode. It is observed that intermediate to high n ballooning modes are in general destabilised by the applied MP field, while external peeling-ballooning modes reorganise to an internal ballooning structure. In addition, extrema in the growth rate spectrum, due to low n kink modes, are observed to be strongly destabilised as predicted by perturbation theory. This work provides proof of principle examination of the 3D peeling-ballooning instability as well as a framework for the optimisation of MP coil configuration

Topics in Magnetohydrodynamics

To understand plasma physics intuitively one need to master the MHD behaviors. As sciences advance, gap between published textbooks and cutting-edge researches gradually develops. Connection from textbook knowledge to up-to-dated research results can often be tough. Review articles can help. This book contains eight topical review papers on MHD. For magnetically confined fusion one can find toroidal MHD theory for tokamaks, magnetic relaxation process in spheromaks, and the formation and stability of field-reversed configuration. In space plasma physics one can get solar spicules and X-ray jets physics, as well as general sub-fluid theory. For numerical methods one can find the implicit numerical methods for resistive MHD and the boundary control formalism. For low temperature plasma physics one can read theory for Newtonian and non-Newtonian fluids etc