Hamilton-Jacobi theory for connecting equilibrium magnetic fields across a toroidal surface supporting a plasma pressure discontinuity

Abstract

This thesis explores the conditions for connecting magnetic fields separated by an infinitely thin current sheet in a plasma. A condition of force balance strongly constrains the allowable continuations of the magnetic field across the surface. As this study is principally motivated by the need to calculate plasma equilibria in non-axisymmetric (3-D) toroidal fusion devices, the current sheet is assumed to be a 2-torus, and there is a focus on current sheets supporting a jump in pressure. Such surfaces serve as interfaces separating volumes of constant-pressure Taylor relaxation in MRXMHD. The topology of the problem allows the adaptation of dynamical systems criteria for the existence of invariant tori in Hamiltonian phase space to infer a necessary criterion for valid continuation. To this end a Hamiltonian-Jacobi equation is derived and termed the pressure jump Hamiltonian system. Given the metric of the toroidal surface, two approaches to applying the Hamilton-Jacobi method are adopted: (a) a continuation approach, in which the magnetic vector field is taken to be known on one side of the interface; and (b) an inverse problem approach, in which the magnetic vector fields on both sides of the interface are taken to be initially unknown, only scalar data on the fluctuation of the square of the field being given. An appeal to the Birkhoff theorem resolves the question as to whether Hamiltonian trajectories map homeomorphically to actual magnetic field lines. A corollary of this reconciliation implies that the winding number of the Hamiltonian orbit is indeed conserved under the mapping as the rotational transform of the corresponding field line. Thus for a given pressure jump and desired rotational transform of the magnetic field proposed for continuation or connection, the criterion used for feasibility is that there exists an invariant torus in the phase space of the pressure-jump Hamiltonian at a given energy (pressure jump) with the same winding number as the desired rotational transform. The criterion is applied computationally by solving for orbits and determining if they result in an ergodic covering of an invariant surface using Greene's residue. This thesis also describes a computer code written by the author that solves the pressure jump Hamiltonian and applies tests based on Greene's residue criterion to create robustness plots that visualise regimes of parameters for which connection is allowed. Conditions for connection are explored for three cases: 1) a simplified version of the pressure jump Hamiltonian to better understand the effects of the perturbative variables, 2) prescribed, smooth interfaces typical of the type that would be used as interfaces in the equilibrium code SPEC, and 3) a flux surface extracted from a partially chaotic volume from a SPEC equilibrium. Continuation across rotational transform discontinuities is also investigated. In some cases energy healing is observed in which increasing energy can disallow a connection, then allow it again at a higher energy. This healing implies that solutions to the problem are allowed (or forbidden) in energy "bands". The implications of all results to MRXMHD and SPEC are then articulated.