Hydromagnetic instabilities of Multi-Region relaxed Magnetohydrodynamics for fusion plasma confinement

Abstract

In toroidal magnetic confinement, the assumption that magnetic field lines lie in nested surfaces leads to a significant simplification of the equilibrium magnetic field problem. The existence of a continuous symmetry, such as toroidal asymmetry, guarantees this property. The property can also be imposed as a constraint in fully three-dimensional configurations, and thereby exploited to yield a computationally efficient solver. For the general class of fully three-dimensional configurations however, magnetic field lines do not always lie in nested surfaces, and the field is a fractal blend of flux surfaces, magnetic islands, and chaotic regions. Mathematically, and computationally, this general class is much harder to describe and solve.

To capture this wider solution class, Dewar, Hole et al (Piecewise-Beltrami MHD equilibria, APS meeting, 2006) proposed a MHD model in which the 3D equilibria could be made up of volumes in which the plasma has a constant pressure, separated by interfaces across which the pressure, field and rotational transform can jump. The force balance condition strongly constrains the allowable continuations of the magnetic field across these barriers. They proposed a variational energy principle, in which the total plasma energy (magnetic plus thermal), was minimized subject to the constraint of constant magnetic helicity and magnetic fluxes, assuming arbitrarily thin, ideal-MHD toroidal interfaces act as barriers to thermal and magnetic relaxation. The resultant stepped-pressure equilibria is the extremum of a multi-region, relaxed magnetohydrodynamic (MRxMHD) energy functional that combines elements of ideal MHD and Taylor relaxation. Numerical solutions have been obtained using the stepped-pressure equilibrium code, SPEC (Hudson et al 2012 Phys. Plasmas 19 112502), where the energy-functional is discretized using a mixed Galerkin-Fourier representation for the magnetic vector potential and the plasma geometry.

Although the MRxMHD energy functional is minimised it is not necessarily stable, as interfaces can be unstable to MHD modes and the plasma volume unstable to tearing modes under an infinitesimal ideal interface perturbations. This thesis investigates the both global and localized linear stability of MRxMHD configurations. Our work builds on and the extends earlier work by Hole et al (2007 Nucl. Fusion 47 746), who studied the MRxMHD linear stability in cylindrical geometry. We have derived the theoretical expressions for the variations of total MRxMHD energy functional with respect to Lagrangian displacements of the barrier interfaces, in which its second variation is obtain with assuming the first variation is zero. Our theoretical framework relates this second variation of the energy functional to the so-called generalized Hessian matrix, which we compute in SPEC. The negative and positive eigenvalues of the generalized Hessian matrix predict the stability of an equilibrium.

To address the MRxMHD stability calculations in detail, several verification studies are conducted on simplified test scenarios in cylindrical, tokamak and stellarator magnetic topologies. In a pressureless cylindrical tokamak, SPEC stability results have been compared to those from the M3D-C1 code and the tearing mode criterion for linear ideal and resistive MHD instability, respectively. In toroidal geometry, we have undertaken a systematic comparison between the marginal stability prediction of the SPEC with the ideal MHD stability code packages MISHKA-1 (tokamak) and CAS3D (stellarator). Finally, we have also compared the SPEC predictions to the resistive inner layer models of Coppi, Greene, Johnson (1966 Nucl. Fusion 6 101) and Glasser, Greene, Johnson (The Physics of Fluids 18, 875-888 (1975)). Combined, these verification studies confirm that the linear stability of the MRxMHD model, and its computational implementation in SPEC, describes linear ideal MHD and tearing mode stability.