ORCID Profile
0000-0002-0019-0356
Current Organisation
University of Colorado at Boulder
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Publisher: Elsevier BV
Date: 11-2016
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2012
DOI: 10.1137/110846865
Publisher: AIP Publishing
Date: 12-2021
DOI: 10.1063/5.0066000
Abstract: A formal series transformation to Birkhoff–Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincaré–Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a non-autonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An ex le is given for an axis with constant torsion near a 1:3 resonance.
Publisher: Elsevier BV
Date: 02-2008
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2009
DOI: 10.1137/080728160
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2005
DOI: 10.1137/040612877
Publisher: IOP Publishing
Date: 11-05-2012
Publisher: AIP Publishing
Date: 12-2021
DOI: 10.1063/5.0049361
Abstract: Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation entirely consisting of invariant submanifolds. A compact regular leaf (a flux surface) of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves, including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields: they are presymplectic systems.
Publisher: IOP Publishing
Date: 13-08-2019
Publisher: IOP Publishing
Date: 13-12-1999
Publisher: IOP Publishing
Date: 22-02-2023
Abstract: A simple property of magnetic fields that minimizes bouncing to passing type transitions of guiding center orbits is defined and discussed. This property, called isoprominence, is explored through the framework of a near-axis expansion. It is shown that isoprominent magnetic fields for a toroidal configuration exist to all orders in a formal expansion about a magnetic axis. Some key geometric features of these fields are described.
Publisher: Elsevier BV
Date: 10-2000
Publisher: Elsevier BV
Date: 10-1998
Publisher: IOP Publishing
Date: 07-1998
Publisher: AIP Publishing
Date: 2021
DOI: 10.1063/5.0035175
Abstract: Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay’s converse Kolmogorov–Arnol’d–Moser condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models: a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.
Location: United States of America
No related grants have been discovered for James Meiss.