There are many situations in both laboratory and astrophysical plasmas where violent eruptions can occur. Such dramatic events, with very short time-scales, cannot be explained solely in terms of linear theory. This research project will develop a theoretical understanding of the non-linear mechanisms responsible for such explosive growth. The focus will be on two types of instability that are relevant in laboratory tokamak plasmas, which are confined by magnetic fields to achieve the conditions necessary for fusion. These are called the edge-localised mode (ELM) and the neoclassical tearing mode (NTM). The ELM is a particularly violent event in a tokamak plasma which leads to a massive, sudden ejection of heat and particles from the plasma surface. We have developed a non-linear theory to explain this phenomenon, and in the process have identified a possible link between ELMs in tokamaks and solar eruptions. Our theory predicts that filaments of plasma erupt from the surface, and these have since been observed on many of the world's major fusion tokamak experiments. A first step of this research project will be to provide a computer code to solve the non-linear equation we have derived, which is interesting in itself as it contains a fractional derivative and a finite time singularity. The code will be used to make quantitative predictions that can be compared with experimental data. We still know little about how these filaments can release so much energy (around a megajoule) in such a short time (around 100 microseconds). A major part of the proposed work will be to explore the energy release mechanism, which will require new physics studies, possibly involving reconnection of the magnetic field lines. Understanding this is particularly important for the next step, multi-billion Euro, international tokamak called ITER, which will be constructed in France. There is a major concern that these ELM events could affect the performance of ITER, and could even cause serious damage to its structure. As well as benefits to the fusion community, we also expect the results to shed light on mechanisms for astrophysical eruptions.The tokamak plasma is generally stable to the NTM unless it gets a 'kick' from another instability. An NTM can then be excited. This kick could come from the ELM described above or, more usually, from another type of rapid instability in the plasma core called the 'sawtooth'. The NTM causes magnetic field lines to break and reconnect because of filamentary currents in the plasma, to create large coherent structures called magnetic islands . The modified magnetic topology is much less effective at confining heat and particles, which is a concern for ITER. We will adapt our theory for the ELM to explore whether or not it can explain the explosive nature of the sawtooth instability also. We will then study the implications of the model for triggering an NTM. There are two important questions for the NTM:(1) How big is the 'kick' that is provided by the ELM or sawtooth (the 'seed')?(2) How big does the kick have to be to trigger an NTM (the 'threshold')?We shall address the first through our improved understanding of ELMs and sawteeth. To answer the second question we will explore how the magnetic islands interact with fine-scale phenomena (such as the particle orbits or plasma turbulence) that influence the transport of pressure and momentum in the plasma. These transport processes influence the filamentary currents that give rise to the NTM. In fact, we believe that under certain conditions they may heal small magnetic islands, providing a threshold for NTM growth. We shall explore the mechanisms which govern this by constructing a new, state-of-the-art computer code. With this code, supported by analytic solutions to simplified model equations, we shall shed new light on reconnection events in plasmas in general, and the NTM in particular.

In a tokamak, the conditions for fusion energy are achieved by confining a hot plasma using a toroidal configuration of magnetic field. Thus, the magnetic field lines lie on a set of toroidal flux surfaces that are nested like a set of Russian dolls. All magnetic field lines on a given flux surface are usually equivalent and, specifically, all carry the same current. However, under certain situations this state can bifurcate to one where some field lines carry more current than others. This filamentation of the current density effectively tears the flux surface apart, creating a chain of so-called magnetic islands. The instability responsible for this is called a tearing mode. Such islands are detrimental to confinement, and therefore it is important to understand the physics of tearing modes. A particularly problematic instability is called the neoclassical tearing mode, or NTM. Small current filaments initially create small so-called "seed" islands. These seed islands reinforce the current filamentation, resulting in a positive feedback mechanism that causes the magnetic islands to grow extremely large. The degradation in confinement causes a drop in the core plasma pressure and a consequent loss in fusion power in a tokamak like ITER. However, this amplification mechanism is only observed when the initial seed island width exceeds a certain threshold of a few centimetres. Although we have ideas for the physics mechanisms that lie behind this threshold, there is no predictive quantitative model. This is largely because for small islands, the distribution of ions in both real and velocity space is important - a 6-dimensional problem. We have developed an expansion in the small ratio of the island width to system size that has enabled us to reduce the system to 4-dimensions - two spatial and two velocity components. Our initial studies indicate that this problem is tractable on modern high end computers, providing a predictive capability for the threshold for neoclassical tearing modes - a key ingredient for specifying the NTM control system on ITER, for example. In a second application of the theory, we are interested in a situation where the magnetic islands are induced by the tokamak operator. This is achieved by applying so-called "resonant magnetic perturbations", or RMPs, to the plasma using a set of current-carrying coils. The motivation for this is to provide a control system for a repetitive sequence of tokamak plasma eruptions, called edge-localised modes, or ELMs. In an ELM, large filaments of plasma erupt from the surface in an event that is reminiscent of solar flares. We believe that these are driven by steep pressure gradients that form near the plasma edge. By driving small magnetic islands in this steep pressure gradient region with RMPs, it is expected that the pressure gradient can be reduced in a controlled way to just below that necessary to trigger an ELM. This is key for ITER, where uncontrolled ELMs will cause excessive erosion of its components at full fusion power. While the technique works on some tokamaks, it does not work on others. To understand this, we need improved models for how the plasma responds to magnetic islands that are driven externally - will it amplify them, as in the case of the NTM, or heal them? This understanding will help specify the ELM control system on ITER. We will develop a new high end computing code to calculate the kinetic plasma response to both natural and driven magnetic islands, using the model we have derived by an analytic reduction of the so-called drift-kinetic theory. Knowledge of the plasma response will enable us to quantify the current filamentation, and hence identify the conditions for which the plasma tends to amplify magnetic islands and when it heals them. We will work with experimentalists to design tests for our predictions against data from today's tokamaks, and make predictions for the requirements of the instability control systems on ITER.