Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields

A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly and thus preserves the non-canonical symplectic form. The algorithm itself is only quasi-geometric due to a series expansion in the orbit parameter. For practical purposes an expansion to the fourth order retains geometric properties down to computer accuracy in typical examples. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates stable long-term orbit dynamics conserving invariants. In Monte Carlo evaluation of transport coefficients, the computational efficiency of quasi-geometric integration is an order of magnitude higher than with a standard fourth order Runge-Kutta integrator.Comment: 38 pages, 11 figure

Energetic ion dynamics and confinement in 3D saturated MHD configurations

In the following theoretical and numerically oriented work, a number of findings have been assembled. The newly devised VENUS-LEVIS code, designed to accurately solve the motion of energetic particles in the presence of 3D magnetic fields, relies on a non-canonical general coordinate Lagrangian formulation of the guiding-centre and full-orbit equations of motion. VENUS-LEVIS can switch between guiding-centre and full-orbit equations with minimal discrepancy at first order in Larmor radius by verifying the perpendicular variation of magnetic vector field, not only including gradients and curvature terms but also parallel currents and the shearing of field-lines. By virtue of a Fourier representation of the fields in poloidal and toroidal coordinates and a cubic spline in the radial variable, the order of the Runge-Kutta integrating scheme is preserved and convergence of Hamiltonian properties is obtained. This interpolation scheme is crucial to compute orbits over slowing-down times, as well as to mitigate the singularity of the magnetic axis in toroidal flux coordinate systems. Three-dimensional saturated MHD states are associated with many tokamak phenomena including snakes and LLMs in spherical or more conventional tokamaks, and are inherent to stellarator devices. The VMEC equilibrium code conveniently reproduces such 3D magnetic configurations. Slowing-down simulations of energetic ions from NBI predict off-axis deposition of particles during LLM MHD activity in hybrid-like plasmas of the MAST. Co-passing particles helically align in the opposite side of the plasma deformation, whereas counter-passing and trapped particles are less affected by the presence of a helical core. Qualitative agreement is found against experimental measurements of the neutron emission. Two opposing approaches to include RMPs in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, embedding the RMPs in a 3D saturated ideal MHD state but excluding stochastic field-lines within the last closed flux-surface. Simulations of fast ion populations from NBI are applied to MAST n=3 RMP coil configuration with 4 different activation patterns. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines, whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma as well as drift resonances

Study of variational symplectic algorithms for the numerical integration of guiding center equations of motion

Questa tesi presenta una discussione dei più moderni algoritmi simplettici variazionali per l'integrazione delle equazioni del moto del centro di guida in particelle cariche in campi magnetici arbitrari statici, utili nello studio di plasmi debolmente collisionali. Differenti varianti degli algoritmi sono presentate, insieme a studi numerici ed analitici che ne evidenziano la stabilità numerica, o la relativa mancanza di essa

Determination of the nature of radial transport in quasi-poloidal stellarator configurations

Mención Internacional en el título de doctorNuclear fusion is one of the most promising solutions to the long-term energy needs of the world. Nevertheless, bringing the source of energy of stars to Earth is not easy. From the different options explored to produce fusion, magnetic confinement is the most developed one and, probably, the first that will be available. Tokamaks and Stellarators are the two most important configuration concepts of this kind, both having a toroidal shape. The main problem magnetic confinement fusion suffers is that all configurations have important losses of energy and particles along the radial direction that makes achieving the required conditions a challenge. Traditionally, those losses have been modelled using neoclassical and turbulent descriptions that assume the existence of an underlying transport of diffusive characteristics. As a result, effective transport coefficients (diffusivities, viscosities, conductivities, etc.) have been estimated to describe the transport processes inside the plasmas confined in these magnetic configurations. Recently, it has been however suggested that there are several important regimes in these devices in which such an assumption may be wrong. As a result, these diffusive-like models may importantly misrepresent the transport dynamics and compromise the performance predictions of larger devices. Among the situations identified where the nature of the radial transport may be fundamentally non-diffusive, there are two particularly meaningful for magnetic confinement devices (see Ref. [1] for a review). The first one is the case of near-marginal transport, in which the plasma profiles (for pressure, temperature, etc) wander locally very close to the thresholds for the excitation of instabilities. In such cases, radial avalanching may become the dominant form of transport, instead of diffusion [2, 3]. In next-generation tokamaks, such as ITER [4], predictions have been made for an almost near-marginal operation in some profiles, due to the fact that turbulent fl uxes scale with a large power of the plasma temperature. Thus, at the much hotter plasmas expected in ITER, this might certainly be an issue to consider. Another example, closer to what we are going to study in this thesis, is the case of radial transport across strong, radially-sheared zonal flows, as shown recently in tokamaks [5, 6, 7]. The problem studied in this thesis, however, refers to transport in stellarators, not tokamaks. Stellarators have seen a recent revival by improving the confinement properties of neoclassical guiding centre orbits by endowing the confining magnetic field with a hidden symmetry usually referred to as quasi-symmetry. Several types of quasi-symmetries exist. The most important ones are quasi-poloidal, quasi-helical and quasiaxisymmetric. We will discuss them in detail in later chapters but, for now, it suffices with saying that quasi-symmetric configurations have a better neoclassical confinement compared to that of standard stellarators. Experimental results from the HSX (helically quasi-symmetric) stellarator [8] have already provided evidence supporting an improved neoclassical confinement [9]. They also have smaller viscosities in the direction of the symmetry, which should in principle facilitate an easier excitation of flows, either by the turbulence itself or externally. Experimental evidence supporting this reduction is also available from HSX [10]. In this context, it is therefore a natural question to ask whether the reduction of losses and better confinement in quasi-symmetric configurations are a mere reduction of turbulent transport levels, or whether there is something more fundamental being changed. The investigation of the latter is where this thesis is centered, focusing in particular on quasi-poloidal configurations. The reduction of the neoclassical poloidal viscosity expected for poloidally quasi-symmetric configuration should facilitate the self-generation of poloidal zonal flows, which are particularly important in terms of affecting radial transport [11]. From the previously mentioned tokamak evidence, it is therefore expected that nondiffusive features of transport might appear more strongly in poloidal quasi-symmetric configurations. Thus, the present thesis investigates whether this is the case or not. Or, more precisely, we will quantify the changes in the nature of radial turbulent transport and attempt to establish whether these changes are (or not) correlated to the level of quasi-poloidal symmetry of the configuration. In order to do it, many gyrokinetic turbulent simulations have been carried out, in a selected configuration with quasi-poloidal symmetry, using the Gene [12] gyrokinetic code (see Chapter 2). The degree of quasi-symmetry of the selected configuration varies, however, strongly with radius. We have used this to our advantage by carrying out local simulations around different radial locations of the same configuration, which has yielded the plethora of data with which the comparative study previously described has been carried out. The characterization of the nature or turbulent transport has been done by means of a methodology that employs tracked particles. These particles may be massless (i.e., tracers) or possess mass and charge. Either way, these particles are tracked as they are advected by the underlying turbulence (previously calculated by Gene ) and, if massive, the different magnetic and parallel drifts that might be present. The temporal dispersion of an initial population of these particles can be used to determine the nature of radial transport rather easily, as we discuss in Chapter 3. However, advecting tracked particles within the advance loop of modern Vlasov gyrokinetic codes is very ine cient and highly unpractical. Gyrokinetic codes have high complexity, strong parallelization and a extremely delicate internal balance. For that reason, we have developed a new and independent tracking code, TRACER, that we have used to carry out all the studies in this thesis. The inner details of this new code are discussed at length in Chapter 4. The discussion of the results of the comparative study previously mentioned is eshed out in Chapter 5. The main conclusion we have drawn is that there is indeed a correlation between the level of quasisymmetry and the nature of radial transport, which becomes more subdi ffusive the larger the level of quasi-symmetry is. The nature of this change is also shown to be connected with the larger ability of the quasisymmetric plasma to excite poloidal flows with strong radial shear, which is very reminiscent of what is found in tokamaks [7]. We have carried out this study both for tracers and massive ions, and very similar results are found in the long-term limit, which makes us believe that the conclusions of this thesis are of importance for the confinement of the thermally confined plasma. The main results of this thesis have been presented in several international conferences and workshops, and have also seen publication in the international journal Physics of Plasmas. A complete list of these publications and presentations can be found in Appendix C. As a last note it is worth saying that, throughout the document, most of the variables will be expressed in Gene units. A very few variables, mostly related with the description of the QPS-configuration, will be however expressed in the International System of Units. The abbreviations used in the document are always introduced and they are part of the general terminology used by the fusion community.Programa Oficial de Doctorado en Plasmas y Fusión NuclearPresidente: Edilberto Sánchez González.- Secretario: José Ramón Martin Solis.- Vocal: Guilhem Dif-Pradalie

THREE-DIMENSIONAL EFFECTS OF ELECTROMAGNETIC FIELDS IN TOKAMAK PLASMAS

The problem of the energy harvesting to face the more and more increasing energy demand is currently challenging. The higher part of our electrical energy (about 80%) is produced by thermoelectrical power plants, which exploit the so-called Non-renewable energy resources (e.g. oil and gas), whose re-growth rate lasts millions of years and are so to be considered as in a fixed amount. On the other hand, the Renewable energy resources are not reduced by their exploitation. For instance, solar and wind energy are obviously both permanent renewable resources, because the energy flow is lower than the energy storage, contrary to the oil resource, where the flow exceeds its natural re-growth rate. Recalling that the renewable energy resources are not able to cover the energy needs (they are often used for the Peak Shaving and not to cover the basis energy demand), it is clear that a new energy resource is necessary to meet the increased energy demand. Moreover, it has to be non-polluting, renewable and continuously available with no interruptions (unlike solar and wind energy, which are affected by the presence of sunlight and wind). This new energy source can be the Nuclear Fusion Energy, a new kind of energy resource that exploits the energy released by the collision and the fusion of two light atoms (such as hydrogen or its isotopes), according to Einstein equation and the mass-energy balance. Although controlled fusion is extremely technologically challenging, a fusion power plant would offer significant advantages over the existing renewable and non-renewable energy sources, such as the practically infinite fuel supply, the absence or air pollution or greenhouses gas during normal operations and the absence of the risk of a nuclear meltdown. The collision of two nuclei can occur if and only if their kinetic energy is high enough to overcome the energy barrier opposing the fusion reaction, due to the long-range Coulomb repulsion. Therefore, the hydrogen gas is heated up to very high temperatures (one hundred million degrees and even more), reaching the Plasma state. Because of this temperature range, the plasma must be confined and must not touch any structure, in order to avoid yielding heat loads as well as mechanical loads. The Tokamak is a fusion machine aimed at the plasma confinement by means of a magnetic field generated by a set of coils surrounding the plasma itself. In principle, the plasma is supposed to be toroidal shaped during normal operations, but this symmetrical condition is ideal, because of many effects which may lead to a non-axisymmetric perturbation of the plasma column. For these reasons, this PhD thesis is devoted to the analysis of some non-axisymmetric plasma perturbations, their effects during the plasma operations and their modelling. The PhD thesis is divided as follows: 1. The first chapter is a brief overview of the main principles the controlled thermonuclear fusion is based on, focusing on the plasma confinement inside a tokamak, the additional heating and the roadmap towards the fusion energy. 2. The second chapter describes the diamagnetic flux evaluation in ITER tokamak for the estimation of the poloidal beta in the presence of non-axisymmetric effects. In particular, the COMPFLUX procedure used for the analysis is presented, then the effects of the main three-dimensional effects are evaluated and the performance of the compensation system is assessed. 3. The third chapter shows the electromechanical effects due to non-axisymmetric halo currents in ITER tokamak. After discussing the mathematical model, the mechanical effects in terms of forces and torques on the structures surrounding the plasma are evaluated. 4. The fourth chapter is devoted to the flux-density field lines tracing and to the identification of non-axisymmetric plasmas. The mathematical model and the procedures developed for the analysis are presented. Afterwards, the standard and geometrical integrators are compared with reference to test cases for which analytical solutions based on the use of Clebsch potentials are available. Finally, the field line tracing technique is used for the non-axisymmetric plasma boundary reconstruction and a novel technique for the 3-D plasma identification is presented and validated. 5. The fifth chapter reports the main conclusions regarding all the topics dealt with this PhD thesis

GENE-3D - ein globaler gyrokinetischer Turbulenzcode für Stellaratoren und gestörte Tokamaks

This thesis describes the development and application of GENE-3D, a global gyrokinetic turbulence HPC code for stellarators. The gyrokinetic equations as well as their implementation and the use of field-aligned coordinates in non-axisymmetric geometries are discussed. GENE-3D is benchmarked for validity and performance. Different geometries of Wendelstein 7-X are investigated for their influence on turbulent properties. Also the influence of the machine size on linear growth rates is studied.Diese Arbeit beschreibt die Entwicklung und Anwendung von GENE-3D, ein globaler gyrokinetischer Turbulenzcode für Stellaratoren. Die gyrokinetischen Gleichungen sowie deren Implementierung und das am Feld ausgerichtete Koordinatensystem werden für nicht-axisymmetrische Geometrien vorgestellt. GENE-3D wird auf Korrektheit getestet.Der Einfluß unterschiedlicher Wendelstein 7-X Geometrien auf den turbulenten Transport und der Einfluß der Maschinengröße auf die linearen Anwachsraten wird untersucht

Design and operation of a harmonic gyrotron based on a cusp electron gun

Strathclyde theses - ask staff. Thesis no. : T13121This thesis presents the results of successful operation of a 2nd harmonic gyrotron based on a cusp electron gun. The numerical and experimental results agreed well with the gyrotron design parameters. Two gyrotrons based on a cusp electron gun were designed: the first gyrotron operated at the 2nd harmonic and the second gyrotron was studied to look at the scaling of this concept for operation at the 7th harmonic at a frequency of 390 GHz. The cusp electron gun was used to produce the electron beam in the gyrotron which was annular in shape. The electron beam had a voltage of 40 kV, a current of 1.5A and a velocity ratio (perpendicular component to horizontal component) of 1.5. The experimental results from the first cusp electron gun and measurements of the high quality electron beam with ~8% velocity spread and ~10% alpha spread are presented. Analytical, numerical and experimental results of a DC harmonic gyrotron are presented. The 3D PIC code MAGIC was used to simulate the interaction of the harmonic gyrotron such as the TE71 mode at the 7th cyclotron harmonic with the large orbit electron beam with the beam thickness and beam spread introduced into the simulation. The interaction cavity of both gyrotrons was in the form of a smooth cylindrical waveguide. The relationship between the cavity dimensions and cavity Q values has been studied for optimized output at the design mode with the aim of suppressing other competing modes. A linear output taper was designed with low mode conversion at the gyrotron output. A Vector Network Analyzer with high frequency millmetre wave heads was used to measure the millimeter wave properties of the gyrotron cavity. Experiments were conducted using the electron gun for the harmonic gyrotron. The gyrotron and electron gun were built as well as the interlock and safety system, pulsed power supply and magnet, the cooling and vacuum system. Millimetre wave radiation was measured for the 2.6 mm diameter cavity gyrotron operating at the 2nd harmonic at a magnetic field of 2.08 T. Experiments demonstrated that the harmonic gyrotron was sensitive to the magnetic field and electron beam parameters. Millimetre wave radiation from 108GHz to 110GHz was measured with the use of a W-band rectifying crystal detector and high pass cut off filters. The frequency of the measured millimeter wave radiation agreed very well with the design and predictions of theory.This thesis presents the results of successful operation of a 2nd harmonic gyrotron based on a cusp electron gun. The numerical and experimental results agreed well with the gyrotron design parameters. Two gyrotrons based on a cusp electron gun were designed: the first gyrotron operated at the 2nd harmonic and the second gyrotron was studied to look at the scaling of this concept for operation at the 7th harmonic at a frequency of 390 GHz. The cusp electron gun was used to produce the electron beam in the gyrotron which was annular in shape. The electron beam had a voltage of 40 kV, a current of 1.5A and a velocity ratio (perpendicular component to horizontal component) of 1.5. The experimental results from the first cusp electron gun and measurements of the high quality electron beam with ~8% velocity spread and ~10% alpha spread are presented. Analytical, numerical and experimental results of a DC harmonic gyrotron are presented. The 3D PIC code MAGIC was used to simulate the interaction of the harmonic gyrotron such as the TE71 mode at the 7th cyclotron harmonic with the large orbit electron beam with the beam thickness and beam spread introduced into the simulation. The interaction cavity of both gyrotrons was in the form of a smooth cylindrical waveguide. The relationship between the cavity dimensions and cavity Q values has been studied for optimized output at the design mode with the aim of suppressing other competing modes. A linear output taper was designed with low mode conversion at the gyrotron output. A Vector Network Analyzer with high frequency millmetre wave heads was used to measure the millimeter wave properties of the gyrotron cavity. Experiments were conducted using the electron gun for the harmonic gyrotron. The gyrotron and electron gun were built as well as the interlock and safety system, pulsed power supply and magnet, the cooling and vacuum system. Millimetre wave radiation was measured for the 2.6 mm diameter cavity gyrotron operating at the 2nd harmonic at a magnetic field of 2.08 T. Experiments demonstrated that the harmonic gyrotron was sensitive to the magnetic field and electron beam parameters. Millimetre wave radiation from 108GHz to 110GHz was measured with the use of a W-band rectifying crystal detector and high pass cut off filters. The frequency of the measured millimeter wave radiation agreed very well with the design and predictions of theory