The motivation of this thesis is to explore the physics of ELMs and the pedestal, an understanding of which is vital to the success of ITER. The ELITE code, which was originally developed to efficiently calculate the edge ideal magneto-hydrodynamic (MHD) stability properties of tokamaks, optimised for the intermediate-high toroidal mode number, n, modes associated with ELMs, has been extended. The motivation was to use the code to explore the stability properties of QH-mode. Further insight to ELM-free scenarios is important for determining suitable ELM mitigated regimes for ITER. Additionally, in 2011 JET finished installing a fully metal ITER-like-wall (ILW) with a beryllium first wall and tungsten divertor. This was performed to demonstrate: firstly to demonstrate the acceptable tritium retention, and secondly the ability to operate a large high power tokamak in within the limitations of the metal wall [67]. Understanding the mechanisms of the pedestal and the resulting ELMs in the presence of the JET-ILW is also highly important, since this will improve predictions of pedestal behaviour in ITER.
Chapter 1. Introduction 1.10. Thesis motivation and outline
Figure 1.7: Time history traces from DIII-D QH-mode discharge 106919 where: (a) plasma current, (b) line-averaged density, (c) product of normalised beta and energy confinement enhancement factor, (d) divertor Dα emission, (e) central ion and electron temperature, (f)
∣ ˙Bθ∣ from magnetic probe, (g) total injected neutral beam power and total radiated power, (h) maximum edge electron pressure gradient and (i) the pedestal electron density. (h) and (i) are determined from mtanh fits to the pedestal n and p from Thomson scattering [58]. Reproduced from [58].
arise and how they couple to form peeling-ballooning modes, which are thought to be the ELM triggering mechanism. The original formalism of the ELITE code is also presented to provide a theoretical basis for the formalism extension. Chapter 3 presents the extension of the ELITE formalism to arbitrary n. This was motivated by the low n dominated phenomena found in QH-mode. Chapter 4 presents benchmarks of the extended ELITE, to show that the formalism is valid. It also presents a diagnostic that has been implemented in the original ELITE version of the code to further explore the drive of the peeling-ballooning instability.
Chapter 5 presents the analysis of a DIII-D QH-mode discharge, which has both types of QH-mode. Applying the new ELITE formalism to the original QH-mode shows that it is able to obtain results that agree with previous work, which is important for code validation. Exploring the low n phenomena of the new QH-mode regime to explain observations is an important step in enhancing the understanding of this new mode of operation, and therefore whether it is applicable to ITER.
Chapter 6 presents the analysis of a database of JET-ILW pulses, which explores the inter-ELM pedestal evolution and compares this to stability analysis. This was a new way of exploring the performance loss that has been experienced since the instal- lation of the JET-ILW. Understanding this is crucial for the success of the upcoming JET D-T campaign, and beyond to ITER.
Chapter 2. Theory and ELITE
Chapter 2
Magnetic Confinement Fusion
Theory and the ELITE code
This chapter provides an introduction to MHD equilibrium and stability, a description of the original ELITE formalism, and an introduction to the Mercier-Luc formalism. Initially, the concept of ideal MHD and its assumptions are presented. Next introduced is the concept of MHD tokamak equilibria, and the equilibrium codes used to obtain equilibria for the results in this thesis, presented in chapters 4, 5 and 6. Next introduced is the concept of the MHD stability and peeling-ballooning (PB) modes. PB modes are thought to be the trigger for ELMs, and are the theoretical underpinning of the ELITE code. Finally the ELITE formalism is presented in its original form, to provide a basis for its extension presented in chapter 3. Also introduced is the Mericer-Luc formalism, which is used as part of the ELITE formalism framework.
2.1
Magnetohydrodynamics and ideal MHD
Magnetohydrodynamics (MHD) is a fluid model that describes the macroscopic equilib- rium and stability properties of a plasma, such that the plasma is a moving conducting fluid in the presence of a magnetic field [68]. It does not require knowledge of the in- dividual particles in the plasma. There are several versions MHD and the most simple version is ideal MHD. Ideal MHD has the following main initial assumptions: the ion gyro radius is zero, the plasma is a single fluid, has no viscosity and has infinite elec- trical conductivity [68]. This is the version of MHD used in this thesis, and therefore the version that will be discussed here. Another type of MHD is resistive MHD, which is defined by finite resistivity. There are also other extended versions of MHD which include two fluid effects, the Hall current term which is neglected in ideal MHD, and also kinetic effects [68]. Despite the simplicity, the ideal MHD equations are still too complex to be solved in their analytical form [68]. The ideal MHD equations are given
by: ∂ρ ∂t + ∇ ⋅ ρv = 0 ρdv dt = J × B − ∇p d dt( p pγ) = 0 E+ v × B = 0 ∇ × E = −∂B∂t ∇ × B = µ0J ∇ ⋅ B = 0 (2.1)
where E is the electric field, B is the magnetic field, J is the current density, ρ is mass density, v is the fluid velocity, p is the pressure, µ0 is the vacuum permeability
constant, γ= 5/3 which is the ratio of specific heats and is adiabatic, and the derivative d/dt = ∂/∂t + v ⋅ ∇ in the first equation is the convective derivative [68]. The first equation is for mass, known as the continuity equation. This equation implies that there is no dissipation of particles from the plasma, which is a reasonable assumption on the MHD time scale [68]. The second term is the momentum equation and describes the momentum of a fluid with three interacting forces: the pressure gradient force, the magnetic force and the inertial force. The third term is the energy equation, and this contains an adiabatic evolution of the plasma. The fourth term is Ohm’s law, which shows the perfect conductivity assumption: that in a reference frame that moves with a plasma the electric field is zero [68]. The final three equations are Maxwell’s equations. Note that the last three equations are the low frequency form of Maxwell’s equa- tions, and the fourth Maxwell’s equation is neglected. This is because deriving the ideal MHD equations requires taking moments of the kinetic equation corresponding to mass, momentum and energy. In the ordering process, further assumptions of ideal MHD are defined which are used to obtain closure of the equations. The first is quasi- neutrality, ni≃ ne, which arises from the assumption that 0∇ ⋅ E can be neglected, so
that locally the electrons respond fast enough to maintain this assumption [68]. This additionally leads to the displacement current being neglected, which is valid since the characteristic thermal ion velocity of the plasma , vT i ≪ c . The second assumption is
that there is no electron inertia in the electron momentum equation, such that me→ 0,
giving electrons an infinitely fast response due to their very small mass [68]. After writing the equations in terms of a single fluid, more information is required to close the system. This is where an assumption of ideal MHD is used: is that the plasma has a high collision rate, making the plasma a collision dominated fluid. This arises from the assumption that higher order moments can be described using Braginskii’s transport theory, which uses basic variables to describe higher order moments [68]. This leads to a set the of closed equations for ideal MHD.