Magnetohydrodynamic Stability in a Levitated Dipole

Plasma confined by a magnetic dipole is stabilized, at low beta, by magnetic compressibility. The ideal magnetohydrodynamic (MHD) requirements for stability against interchange and high-n ballooning modes are derived at arbitrary beta for a fusion grade laboratory plasma confined by a levitated dipole. A high beta MHD equilibrium is found numerically with a pressure profile near marginal stability for interchange modes, a peak local beta of β ∼ 10, and volume averaged beta of β ∼ 0.5. This equilibrium is demonstrated to be ballooning stable on all field lines. The dipole magnetic field is the simplest and most common magnetic field configuration in the universe. It is the magnetic far-field of a single, circular current loop, and it represents the dominate structure of the middle magnetospheres of magnetized planets and neutron stars. The use of a dipole magnetic field generated by a levitated ring to confine a hot plasma for fusion power generation was first considered by Akira Hasegawa [1, 2]. In order to eliminate losses along the field lines Hasegawa suggested the use of a levitated ring. He postulated that if a hot plasma having pressure profiles similar to those observed in nature could be confined by a laboratory dipole magnetic field, this plasma might also be immune to anomalous (outward) transport of plasma energy and particles. The dipole confinement concept is based on the idea of generating pressure profiles near marginal stability for low-frequency magnetic and electrostatic fluctuations. From ideal MHD marginal stability results when the pressure profile satisfies the adiabaticity condition [3, 4], δ(pV ) = 0, where p is plasma pressure, V is the flux tube volume (V ≡ ∮ d /B) and γ = 5/3. This condition leads to dipole pressure profiles that scale with radius as r−20/3, similar to energetic particle pressure profiles observed in the Earth’s magnetosphere. This condition limits the peak pressure i.e. ppeak ≤ pedge(Vedge/Vpeak) and a relatively low pressure at the plasma edge requires a large flux expansion, i.e. Vedge/Vpeak 1. At low beta the magnetic field in the plasma will closely approximate the vacuum field. At finite beta the equilibrium field can be determined from a solution of the Grad-Shafranov equation. At sufficiently high beta the stability of MHD ballooning modes needs to be examined. The high beta MHD stability limit has been examined by several authors [5, 6, 7] in the magnetospheric context. For the magnetospheric problem it is necessary to consider rotation, anisotropy (p⊥ = p‖) as well as the boundary condition where the field lines enter the conducting regions near the planetary poles. Chan et al.[7] utilize a low beta equilibrium expansion in the ballooning calculation and their results are suspect at high beta[8]. As a laboratory approach to controlled fusion a circular magnet that is located within a plasma will generate a dipole configuration. To avoid losses on supports the ring needs to be superconducting and be magnetically levitated within the vacuum chamber. Since a large flux expansion is necessary to obtain a fusion grade plasma (for fixed edge plasma parameters) the configuration tends to require a small coil that is levitated within a relatively large vacuum chamber. An initial test of this concept is embodied in the Levitated Dipole Experiment[9] (LDX) ∗Also at: Department of Applied Physics, Columbia University, New York, NY 10027