Inter-ELM evolution of the edge current density profile on the ASDEX upgrade tokamak

The sudden decrease of plasma stored energy and subsequent power deposition on the first wall of a tokamak due to edge localised modes (ELMs) is potentially detrimental to the success of a future fusion reactor. Understanding and control of ELMs is critical for the longevity of these devices and also to maximise their performance. The commonly accepted picture of ELMs posits a critical pressure gradient and current density in the plasma edge, above which coupled magnetohy drodynamic peeling-ballooning modes become unstable. Much analysis has been presented in recent years on the spatial and temporal evolution of the edge pressure gradient. However, the edge current density has typically been overlooked due to the difficulties in measuring this quantity. In this thesis, a novel method of current density recovery is presented, using the equilibrium solver CLISTE to reconstruct a high resolution equilibrium utilising both external magnetic and internal edge kinetic data measured on the ASDEX Upgrade tokamak. The evolution of the edge current density relative to an ELM crash is presented, showing that a resistive delay in the buildup of the current density is unlikely. An uncertainty analysis shows that the edge current density can be determined with an accuracy consistent with that of the kinetic data used. A comparison with neoclassical theory demonstrates excellent agreement be- tween the current density determined by CLISTE and the calculated profiles. Three ELM mitigation regimes are investigated: Type-II ELMs, ELMs sup- pressed by external magnetic perturbations, and Nitrogen seeded ELMs. In the first two cases, the current density is found to decrease as mitigation on- sets, indicating a more ballooning-like plasma behaviour. In the latter case, the flux surface averaged current density can decrease while the local current density increases, providing a mechanism to suppress both the peeling and ballooning modes.

Two closely coupled lasers, dynamics & applications

The dynamics of two mutually coupled identical single-mode semi-conductor lasers are theoretically investigated. For small separation and large coupling between the lasers, symmetry-broken one-colour states are shown to be stable. In this case the light output of the lasers have significantly different intensities whilst at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable two-colour states, where both single-mode lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. For low coupling but possibly large separation, the frequency of the relaxation oscillations of the freerunning lasers defines the dynamics. Chaotic and quasi-periodic states are identified and shown to be stable. For weak coupling undamped relaxation oscillations dominate where each laser is locked to three or more odd number of colours spaced by the relaxation oscillation frequency. It is shown that the instabilities that lead to these states are directly connected to the two colour mechanism where the change in the number of optical colours due to a change in the plane of oscillation. At initial coupling, in-phase and anti-phase one colour states are shown to emerge from “on” uncoupled lasers using a perturbation method. Similarly symmetry-broken one-colour states come from considering one free-running laser initially “on” and the other laser initially “off”. The mechanism that leads to a bi-stability between in-phase and anti-phase one-colour states is understood. Due to an equivariant phase space symmetry of being able to exchange the identical lasers, a symmetric and symmetry-broken variant of all states mentioned above exists and is shown to be stable. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-colour, symmetric and symmetry-broken two-colour, symmetric and symmetry-broken undamped relaxation oscillations, symmetric and symmetry-broken quasi-periodic, and symmetric and symmetry-broken chaotic states. As symmetry-broken states always exist in pairs, they naturally give rise to bi-stability. Several of these states show multistabilities between symmetric and symmetry-broken variants and among states. Three memory elements on the basis of bi-stabilities in one and two colour states for two coupled single-mode lasers are proposed. The switching performance of selected designs of optical memory elements is studied numerically.

The influence of thermal plasma profiles on low-frequency Alfvén eigenmode dynamics in a tokamak

The confinement of fast particles, present in a tokamak plasma as nuclear fusion products and through external heating, will be essential for any future fusion reactor. Fast particles can be expelled from the plasma through their interaction with Alfvén eigenmode (AE) instabilities. AEs can exist in gaps in the Alfvén continuum created by plasma equilibrium non-uniformities. In the ASDEX Upgrade tokamak, low-frequency modes in the frequency range from f ≈ 10 − 90kHz, including beta-induced Alfvén eigenmodes (BAEs) and lower frequency modes with mixed Alfvén and acoustic polarisations, have been observed. These exist in gaps in the Alfvén continuum opened up by geodesic curvature and finite plasma compressibility. In this thesis, a kinetic dispersion relation is solved numerically to investigate the influence of thermal plasma profiles on the evolution of these low-frequency modes during the sawtooth cycle. Using information gained from various experimental sources to constrain the equilibrium reconstructions, realistic safety factor profiles are obtained for the analysis using the CLISTE code. The results for the continuum accumulation point evolution are then compared with experimental results from ASDEX Upgrade during periods of ICRH only as well as for periods with both ICRH and ECRH applied simultaneously. It is found that the diamagnetic frequency plays an important role in influencing the dynamics of BAEs and low-frequency acoustic Alfvén eigenmodes, primarily through the presence of gradients in the thermal plasma profiles. Different types of modes that are observed during discharges heated almost exclusively by ECRH were also investigated. These include electron internal transport barrier (eITB) driven modes, which are observed to coincide with the occurrence of an eITB in the plasma during the low-density phase of the discharge. Also observed are BAE-like modes and edge-TAEs, both of which occur during the H-mode phase of the discharge.

Mathematical modelling of the solidification of liquid tin with electromagnetic stirring

It is well known that during alloy solidification, convection currents close to the so-lidification front have an influence on the structure of dendrites, the local solute concentration, the pattern of solid segregation, and eventually the microstructure of the casting and hence its mechanical properties. Controlled stirring of the melt in continuous casting or in ingot solidification is thought to have a beneficial effect. Free convection currents occur naturally due to temperature differences in the melt and for any given configuration, their strength is a function of the degree of superheat present. A more controlled forced convection current can be induced using electro-magnetic stirring. The authors have applied their Control-Volume based MHD method [1, 2] to the problem of tin solidification in an annular crucible with a water-cooled inner wall and a resistance heated outer one, for both free and forced convection situations and for various degrees of superheat. This problem was studied experimentally by Vives and Perry [3] who obtained temperature measurements, front positions and maps of electro-magnetic body force for a range of superheat values. The results of the mathematical model are compared critically against the experimental ones, in order to validate the model and also to demonstrate the usefulness of the coupled solution technique followed, as a predictive tool and a design aid. Figs 6, refs 19.

Liquid metal turbulent flow dynamics in a cylindrical container with free surface: experiment and numerical analysis

The liquid metal flow in induction crucible models is known to be unstable, turbulent and difficult to predict in the regime of medium frequencies when the electromagnetic skin-layer is of considerable extent. We present long term turbulent flow measurements by a permanent magnet incorporated potential difference velocity probe in a cylindrical container filled with eutectic melt In-Ga-Sn. The parallel numerical simulation of the long time scale development of the turbulent average flow is presented. The numerical flow model uses an implicit pseudo-spectral code and k-w turbulence model, which was recently developed for the transitional flow modelling. The results compare reasonably to the experiment and demonstrate the time development of the turbulent flow field and the turbulence energy.

Buoyancy driven flow and its stability in a horizontal rectangular channel with an arbitrary oriented transversal magnetic field

An MHD flow is considered which is relevant to horizontal Bridgman technique for crystal growth from a melt. In the unidirectional parallel flow approximation an analytical solution is found accounting for the finite rectangular cross section of the channel in the case of a vertical magnetic field. Numerical pseudo-spectral solutions are used in the cases of arbitrary magnetic field and gravity vector orientations. The vertical magnetic field (parallel to the gravity) is found to be he most effective to damp the flow, however, complicated flow profiles with "overvelocities" in the comers are typical in the case of a finite cross-section channel. The temperature distribution is shown to be dependent on the flow profile. The linear stability of the flow is investigated by use of the Chebyshev pseudospectral method. For the case of an infinite width channel the transversal rolls instability is investigated, and for the finite cross-section channel the longitudinal rolls instability is considered. The critical Gr number values are computed in the dependence of the Ha number and the wave number or the aspect ratio in the case of finite section.

Magnetic levitation fluid dynamics

This work is concerned with the accurate computation of flow in a rapidly deforming liquid metal droplet, suspended in an AC magnetic field. Intense flow motion due to the induced electromagnetic force distorts dynamically the droplet envelope, which is initially spherical. The relative positional change between the liquid metal surface and the surrounding coil means that fluid flow and magnetic field computations need to be closely coupled. A spectral technique is used to solve this problem, which is assumed axisymmetric. The computed results are compared against a physical experiment and "ideal sphere" analytic solutions. A comparison between the "magnetic pressure" approximation and the full electromagnetic force solutions, shows fundamental differences; the full electromagnetic force solution is necessary for accurate results in most practical applications of this technique. The physical reason for the fundamental discrepancy is the difference in the electromagnetic force representation: only the gradient part of the full force is accounted for in the "magnetic pressure" approximation. Figs 9, Refs 13.

Investigation of the cold crucible melting process: experimental and numerical study

The dynamic process of melting different materials in a cold crucible is being studied experimentally with parallel numerical modelling work. The numerical simulation uses a variety of complementing models: finite volume, integral equation and pseudo-spectral methods combined to achieve the accurate description of the dynamic melting process. Results show the temperature history of the melting process with a comparison of the experimental and computed heat losses in the various parts of the equipment. The free surface visual observations are compared to the numerically predicted surface shapes.

Pseudo-spectral solutions for fluid flow and heat transfer in electro-metallurgical applications

The pseudo-spectral solution method offers a flexible and fast alternative to the more usual finite element/volume/difference methods, particularly when the long-time transient behaviour of a system is of interest. Since the exact solution is obtained at the grid collocation points superior accuracy can be achieved on modest grid resolution. Furthermore, the grid can be freely adapted with time and in space, to particular flow conditions or geometric variations. This is especially advantageous where strongly coupled, time-dependent, multi-physics solutions are investigated. Examples include metallurgical applications involving the interaction of electromagnetic fields and conducting liquids with a free sutface. The electromagnetic field then determines the instantaneous liquid volume shape and the liquid shape affects in turn the electromagnetic field. In AC applications a thin "skin effect" region results on the free surface that dominates grid requirements. Infinitesimally thin boundary cells can be introduced using Chebyshev polynomial expansions without detriment to the numerical accuracy. This paper presents a general methodology of the pseudo-spectral approach and outlines the solution procedures used. Several instructive example applications are given: the aluminium electrolysis MHD problem, induction melting and stirring and the dynamics of magnetically levitated droplets in AC and DC fields. Comparisons to available analytical solutions and to experimental measurements will be discussed.

Pseudo-spectral solutions for fluid flow and heat transfer in electro-metallurgical applications

The pseudo-spectral solution method offers a flexible and fast alternative to the more usual finite element and volume methods, particularly when the long-time transient behaviour of a system is of interest. The exact solution is obtained at grid collocation points leading to superior accuracy on modest grids. Furthermore, the grid can be freely adapted in time and space to particular flow conditions or geometric variations, especially useful where strongly coupled, time-dependent, multi-physics solutions are investigated. Examples include metallurgical applications involving the interaction of electromagnetic fields and conducting liquids with a free surface. The electromagnetic field determines the instantaneous liquid volume shape, which then affects the electromagnetic field. A general methodology of the pseudo-spectral approach is presented, with several instructive example applications: the aluminium electrolysis MHD problem, induction melting in a cold crucible and the dynamics of AC/DC magnetically levitated droplets. Finally, comparisons with available analytical solutions and to experimental measurements are discussed.

Levitated droplet oscillations: effect of internal flow

The numerical model for electrically conducting liquid droplets levitated in AC magnetic field is extended to demonstrate various factors affecting the accuracy of material property value measurements. The effects included are the electromagnetic force induced stirring and the resulting turbulence, thermo-capillary convection, and the droplet rotation. The results are validated against available analytical solutions.

Effects of magnetic fields on crystal growth

The effects of a constant uniform magnetic field on dendritic solidification were investigated using a 2-dimensional enthalpy based numerical model. The interaction between thermoelectic currents and the magnetic field generates a Lorentz force that creates a flow. This flow causes a change in the morphology of the dendrite; secondary growth is promoted on one side of the dendrite arm and the tip velocity of the primary arm is increased.

Shallow water model for aluminium electrolysis cells with variable top and bottom

The MHD wave instability in commercial cells for electrolytic aluminium production is often described using ‘shallow water’ models. The model [1] is extended for a variable height cathode bottom and anode top to account for realistic cell features. The variable depth of the two fluid layers affects the horizontal current density, the wave development and the stability threshold. Instructive examples for the 500 kA cell are presented.