Anderson-localized ballooning modes in general toroidal plasmas

Abstract

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...

Anderson localization of ballooning modes, quantum chaos and the stability of compact quasiaxially symmetric stellarators

“…These findings are similar to those from the ballooning stability analysis 9 for H-1NF, while the ten field period stellarator ballooning stability datacube 10 showed stronger dependence on θ k and s, but weak dependence on α. While localized structures have been found for the ten field period stellarator and the H1-NF heliac, it is somewhat surprising to find them also in the QAS because this is supposed to have tokamak-like properties.…”

“…Eigenfunction localization is seen to increase near the plasma edge where the magnetic shear is weak or zero. Similar low-shear localization of the eigenfunction was also found by Dewar and Cuthbert 9 for the H-1NF heliac.…”

“…The method makes use of the additional geometric and profile information contained in the results of a large set of local ballooning eigenvalue calculations to infer global mode stability. Ballooning eigenvalue isosurfaces have been found which exhibit radial, poloidal and toroidal localization 9 for the H-1NF heliac 11 at The Australian National University and for ten field period stellarators 10,12 related to the Large Helical Device (LHD) 13 in Japan. Given these isosurfaces, ray tracing can be used to predict the occurrence of kinetic stabilization of β for a plasma configuration.…”

“…In this paper we report on the application to the QAS of a well known approach 8 to investigating ballooning mode spectra which has recently been applied to stellarators 9,10 , enhanced by new high performance computing and visualization tools. The method makes use of the additional geometric and profile information contained in the results of a large set of local ballooning eigenvalue calculations to infer global mode stability.…”

Strong “Quantum” Chaos in the Global Ballooning Mode Spectrum of Three-Dimensional Plasmas

“…According to the perturbation expansion in q ′ described in [22], a quadratic form in α, θ k should form a good approximation to λ − λ min (q) in the neighborhood of the central branch. Accordingly a least-squares fit on each surface was performed to provide a simple analytical description of the (0, 0) [22] branch.…”