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

Abstract

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...

Statistical characterization of the interchange-instability spectrum of a separable ideal-magnetohydrodynamic model system

“…These devices are called three-dimensional because they possess no continuous geometrical symmetries, and thus there is no separation of variables to reduce the dimensionality of the eigenvalue problem. It has been shown [18] that the semiclassical limit (a Hamiltonian ray tracing problem) for ballooning instabilities in such geometries may be strongly chaotic because there are no ignorable coordinates in the ray Hamiltonian.…”

MHD Ballooning Instability in the Plasma Sheet

“…Thus, further approximations must be made to simplify these two equations. Fortunately a so-called WKB-ballooning formalism has been developed to simplify these equations by taking advantage of the nature of solutions that the perpendicular wavelength is much shorther than the parallel wavelength [Dewar and Glasser, 1983;Nevins and Pearlstein, 1988;Dewar et al, 2001]. Adopting the WKB-ballooning formalism, we consider the eikonal representation of the perturbed quantities, Φ = ie iSΦ , where S 1 is the WKB eikonal and B · ∇S = 0.…”

Influence of pressure-gradient and shear on ballooning stability in stellarators

“…Before proceeding, we should note that the lack of a continuous symmetry in devices such as the stellarator greatly complicates the extension from the infinite-n, localized ballooning modes to the finite-n, global modes [7,8]. For the axi-symmetric tokamak, it is a relatively straight-forward procedure to construct global modes from the localized modes [9].…”