Matthew Hole

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.

Integrated plasma physics modelling for the Culham steady state spherical tokamak fusion power plant

Integrated modelling of important plasma physics issues related to the design of a steady-state spherical tokamak (ST) fusion power plant is described. The key is a steady-state current drive, and 92% of this is provided by a combination of bootstrap and diamagnetic currents, both of which have a substantial toroidal component in a ST. The remaining current is to be provided by either neutral beam injection or radio-frequency waves, and various schemes for providing this are discussed and quantified. The desire to achieve a high bootstrap current drives the design to high plasma pressure, ? (normalized to the magnetic field pressure), and high elongation. Both these requirements have implications for ideal magneto-hydrodynamic instability which are discussed. Confinement is addressed both through comparison with the recent scaling laws developed from the conventional tokamak database and self-consistent one-dimensional modelling of the transport processes. This modelling shows that the power required for the current drive (~50?MW) is sufficient to heat the plasma to a regime where more than 3?GW of fusion power is produced, taking into account the dilution due to He ash and prompt ?-particle losses, which are small. A preliminary study of the micro-instabilities, which may be responsible for the turbulent transport is provided. Given assumptions about the particle confinement, we make estimates of the fuelling requirements to maintain the steady state. Finally, the power loading due to the exhaust is derived using theory-based scalings for the scrape-off layer width.

ELM Characteristics in MAST

Edge localized mode (ELM) characteristics in a large spherical tokamak (ST) with significant auxiliary heating are explored. High confinement is achieved in mega ampere spherical tokamak (MAST) at low ELM frequencies even though the ELMs exhibit many type III characteristics. These ELMs are associated with a reduction in the pedestal density but no significant change in the pedestal temperature or temperature profile, indicating that energy is convected from the pedestal region into the scrape-off layer. Power to the targets during an ELM arrives predominantly at the low field outboard side. ELM effluxes are observed up to 20 cm from the plasma edge at the outboard mid-plane and are associated with the radial motion of a feature at an average velocity of 0.75 km s−1. The target balance observed in MAST is potentially rather favourable for the ST since H-mode access is facilitated in a regime where ELM losses flow mostly to the large wetted area, outboard targets and, in addition, the target heat loads are reduced by an even distribution of power between the upper and lower targets.

Overview of recent experimental results on MAST

Note: Proc. 19th IAEA Fusion Energy Conference, Lyon, France, October 2002, IAEA-CN-94/EX/OV2-3 Reference CRPP-CONF-2002-068 Record created on 2008-05-13, modified on 2017-05-12

Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem

This work was supported in part by the U.S. Department of Energy Contract No. DE-AC02-76CH03073 and Grant No. DE-FG02-99ER54546 and the Australian Research Council.

Equilibria and stability in partially relaxed plasma–vacuum systems

We develop a multiple interface variational model, comprising multiple Taylor-relaxed plasma regions separated by ideal MHD barriers. The magnetic field in each region is Beltrami, ∇× B = µB, and the pressure constant. Between regions the pressure, field strength, and rotational transform may have step changes at the ideal barrier. A principle motivation is the development of a mathematically rigorous ideal MHD model to describe intrinsically 3D equilibria, with nonzero internal pressure, using robust KAM surfaces as the barriers. This article chiefly addresses whether the stability of two interface configurations with continuous rotational transform, but vanishing interface separation, is different from the stability of a single interface configuration with jump in the rotational transform. To make the problem analytically tractable, we derive the equilibria and stability of a multi-interface plasma in a periodic cylinder, generalizing the cylindrical treatment of Kaiser and Uecker (2004 Q. J. Mech. Appl. Math. 57 1–17). For two interfaces with no jump in rotational transform, we show that one eigenmode has in-phase interface displacements, and an eigenvalue that converges to the single barrier case in the limit of vanishing interface width. The complementary eigenmode is out-of-phase, and highly unstable. Physically, the unstable eigenmode is driven by the parallel current, and caused by the high shear required to match the different rotational transform on each interface. In the limit that the interface separation vanishes, the shear and parallel current density become infinite, and the parallel current between the interfaces nonzero. Surfaces with out-of-phase displacements will then collide, unless the amplitude goes to zero as the interface separation goes to zero. These results suggest the hypothesis that KAM barriers with different irrational rotational transform on either side may be allowable without violating nonlinear stability.

Identifying the impact of rotation, anisotropy, and energetic particle physics in tokamaks

In this paper we study the effects of poloidal and toroidal rotation, and anisotropy in tokamaks. To resolve these effects from uncertainties in the data, we introduce a Bayesian inference framework which calculates the magnetic configuration probabilistically using motional Stark effect and magnetic data. Drawing on these calculations, we compute the poloidal and toroidal Mach numbers in MAST for a discharge with good rotation data. Our calculations confirm that the poloidal Mach number Ms,θ = vθ/vi × B/Bθ is near zero (with vθ and vi the poloidal and thermal velocity, respectively), even on the outboard side where the scaling of poloidal field strength Bθ to total field B is large. In contrast, the toroidal rotation of this plasma reaches a Mach number of 0.5 on axis. The impact of the toroidal rotation on the equilibrium reconstructions of the plasma is however small: it acts to increase the radius of the magnetic axis by ≈1%, and lower the central safety factor by ≈5%. In comparison, corrections to make the pressure profile consistent with internal measurements such as charge exchange recombination spectroscopy and Thomson scattering have a much larger impact. In other work we compute the level of anisotropy from a TRANSP simulation of a neutral beam-heated MAST discharge. This shows a large level of anisotropy, with p⊥/p∥ ≈ 1.7, sufficient to boost the central safety factor by 15%. For this discharge, which is representative of many MAST discharges, the effect of anisotropy and consistent pressure profiles is more pronounced than the toroidal rotation of the plasma.

Compressional Alfven Eigenmodes on MAST

Magnetic fluctuations at frequencies ω ωci driven by neutral-beam injection heating and identified as compressional Alfven eigenmodes (CAEs) have been observed on MAST. The measured toroidal mode numbers are in the range 4 < |n| < 10 and waves rotate in both co- and counter-current directions. The frequency variation is consistent with an Alfvenic scaling, and modes are elliptically polarized with a significant magnetic field component aligned parallel to the equilibrium field. Frequency clustering of modes occurs on three frequency scales. At the finest scale there are multiple modes each separated by a constant frequency ~10–20 kHz; this is shown to be a result of modulation by low-frequency tearing modes. A larger scale frequency splitting exists in the range 100–150 kHz; these have consecutive toroidal mode numbers and are in agreement with numerical modelling. Finally, modes exist at frequencies close to ω = ωci and ωci/2 consistently with previous observations on START and DIII-D suggesting that the CAEs exist in two distinct ranges of k∥. Calculations of CAEs suggest that the modes are localized at r/a ~ 0.5. The modes form within a potential well due to the variation of (nq/κρ)2, and are not directly influenced by variations in vA. This is distinct from observations based on ion cyclotron emission in conventional aspect ratio tokamaks which indicate that CAE modes occur closer to the plasma edge and that their existence relies on a competition between k⊥ and 1/vA.

Relaxed plasma equilibria and entropy-related plasma self-organization principles

The concept of plasma relaxation as a constrained energy minimization is reviewed. Recent work by the authors on generalizing this approach to partially relaxed threedimensional plasma systems in a way consistent with chaos theory is discussed, with a view to clarifying the thermodynamic aspects of the variational approach used. Other entropy-related approaches to finding long-time steady states of turbulent or chaotic plasma systems are also briefly reviewed.

Stepped Pressure Profile Equilibria in Cylindrical Plasmas via Partial Taylor Relaxation

We develop a multiple interface variational model, comprising multiple Taylor-relaxed plasma regions separated by ideal magnetohydrodynamic (MHD) barriers. A principal motivation is the development of a mathematically rigorous ideal MHD model to describe intrinsically three-dimensional equilibria, with non-zero internal pressure. A second application is the description of transport harriers as constrained minimum energy states. As a first example, we calculate the plasma solution in a periodic cylinder, generalizing the analysis of the treatment of Kaiser and Uecker (2004 Q. J. Mech, Appl. Math. 57. 1-17). who treated the single interface in cylindrical geometry, Expressions for the equilibrium field are generated, and equilibrium states computed. Unlike other Taylor relaxed equilibria, for the equilibria investigated here, only the plasma core necessarily has reverse magnetic shear. We show the existence of tokamak-like equilibria, with increasing safety factor and stepped-pressure profiles.