Ballooning instabilities are investigated in three-dimensional magnetic toroidal plasma confinement systems with low global magnetic shear. The lack of any continuous symmetry in the plasma equilibrium can lead to these modes being localized along the field lines by a process similar to Anderson localization. This produces a multibranched local eigenvalue dependence, where each branch corresponds to a different unit cell of the extended covering space in which the eigenfunction peak resides. These phenomena are illustrated numerically for the three-field-period heliac H-1, and contrasted with an axisymmetric s-α tokamak model. The localization allows a perturbative expansion about zero shear, enabling the effects of shear to be investigated. Ballooning instabilities are pressure-driven ideal magnetohydrodynamic (MHD) instabilities which limit the maximum β (plasma pressure/magnetic pressure) that can be obtained in a plasma. They are localized about regions where the field lines are concave to the plasma, which are known as unfavourable regions of curvature. Another localizing influence is the magnetic shear, which measures the rate at which neighboring field lines at different minor radii separate as they wind their way around the torus. Large shear helps stabilize these modes, thereby playing an important role in the MHD stability. In this paper however we consider the effects of very small or zero shear, such as occurs in the heliac class of stellarators or in the shear-reversal layers of an advanced tokamak.We begin by making the usual assumption that the magnetic field lines map out nested flux surfaces, or magnetic surfaces. These are labeled using a normalizedtoroidal-flux variable s, which varies between zero at the center of the plasma and unity at the plasma edge. Within each surface the straight-field-line poloidal θ and toroidal ζ angle variables are defined such that the field lines appear as straight lines in the (θ, ζ) plane. The magnetic field may then be written B = ∇ζ×∇ψ − q∇θ×∇ψ ≡ ∇α×∇ψ, where the field-line label α ≡ ζ − qθ. Here, 2πψ represents the poloidal magnetic flux, while q = q(s) is the safety factor (inverse of rotational transform), which is equal to the average number of toroidal circuits traversed by a field line per poloidal circuit traversed around the torus.Ballooning modes can be characterized as having a long parallel and short perpendicular wavelength with respect to the field lines. By ordering the perpendicular wavelength to be small and expanding to lowest order in an asymptotic series the local mode behavior can be expressed by a one-dimensional equation along a field line [1]. Taking the plasma to be incompressible, the ballooning equation may be written [2]where the eigenfunction ξ is related to the mode displacement while the eigenvalue λ is equal to the mode growth rate squared. This represents the local stability, local to a field line. In forming global modes, ray tracing must be performed in the three-dimensional λ phase space to determine which of these local sol...

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a k-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.PACS numbers: 52.35. Py, 52.55.Hc, 05.45.Mt In design studies for new magnetic confinement devices for fusion plasma experiments (e.g. investigations [1,2] leading to the proposed National Compact Stellarator Experiment, NCSX [3]), the maximum pressure that can stably be confined in any proposed magnetic field configuration is routinely estimated by treating the plasma as an ideal magnetohydrodynamic (MHD) fluid. One linearizes about a sequence of equilibrium states with increasing pressure, and studies the spectrum of normal modes (frequency ω) to determine when there is a component with Im ω > 0, signifying instability.Even with the simplification obtained by using the ideal MHD model, the computational task of determining the theoretical stability of a three-dimensional (i.e. nonaxisymmetric) device, such as NCSX or the four currently operating helical axis stellators [4], remains a challenging one.The problem can be posed as a Lagrangian field theory, with the potential term being the energy functional δW [5]. For a static equilibrium, the kinetic energy is quadratic in ω, so that ω 2 is real. Thus instability occurs when ω 2 < 0. There are two main approaches to analyzing the spectrum-local and global. * Permanent address: Research School of Physical Sciences & Engineering, The Australian National University. E-mail: robert.dewar@anu.edu.au.In the local approach, which is used for analytical simplification, one orders the scale length of variation of the eigenfunction across the magnetic field lines to be short compared with equilibrium scale lengths [6]. Both interchange and ballooning stability can be treated by solving the general ballooning equations [7], a system of ordinary differential equations defined on a given magnetic field line.The global (Galerkin) approach is to expand the plasma displacement field in a finite basis set, inserting this ansatz in the Lagrangian to find a matrix eigenvalue representation of the spectral problem. This approach has been implemented for ideal MHD in threedimensional plasmas in two codes, TERPSICHORE [8] and CAS3D [9].Although the Galerkin approach is potentially exact, if one could use a complete, infinite basis set, it is in practice computationally challenging due to the large number of basis functions required to resolve localized instabilities. This leads to very large matrices which must be diagonalize...

It is shown that the coexistence of toroidally nonlocalized ideal-hydromagnetic ballooning instabilities, with a quasidiscrete spectrum, and toroidally localized ballooning instabilities with a broad continuous spectrum, as predicted by Dewar and Glasser [Phys. Fluids 26, 3038 (1983)] can be realized in a Mercier-unstable equilibrium case modeling the Large Helical Device (LHD) [A. Iiyoshi et al., Fusion Technol. 17, 148 (1990)] with a broad pressure profile. The quasidiscrete, interchange branch corresponds to extended modes that can be understood on the basis of a ripple-averaged ballooning equation, whereas the broad-continuum, ballooning branch corresponds to modes localized along a flux tube. The physical origin of the two branches is discussed.

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