| A plasma flow, confined to a vessel in 3-space, quickly relaxes to a configuration with the least energy compatible with its helicity, a measure of the wrapping and coiling of the orbits of the field around each other. We study these energy-minimizing flows on axisymmetric solid tori representing possible "tokamak" plasma confinement devices, proving that they are first eigenfields of curl with certain boundary conditions. Exploiting the rotational symmetry of the problem, using Rayleigh quotient arguments, and deriving a new formula for the "cross-helicity" of flows on disjoint domains in 3-space, we show that when these fields are axisymmetric, their fundamental topological structure depends very little on the geometry of the vessel. Each of these fields has a positive component in the direction of rotation inside the vessel, and no component in that direction on the boundary. These flows have no stationary points, on the interior or on the boundary. Each is tangent to a family of integral tori, and every torus is smoothly immersed, while most are smoothly embedded. Singular integral surfaces are rare (we conjecture that they do not occur on domains of convex cross-section), and when they do appear their topology is strictly controlled. On the nonsingular integral tori, the orbits of these fields are roughly helical, and the helices are always right-handed. Corresponding results should apply to axisymmetric balls, representing possible "spheromak" devices, and we hope to extend these results to that case in a following paper. |
