Tangential discontinuities and the optical analogy for stationary fields III. Zones of exclusion

Abstract

Abstract This paper presents a number of formal examples of the bifurcation of individual flux surfaces by the pressure maxima imposed by the fields on either side. An approximate necessary and sufficient criterion for the convexity of the pressure maximum is provided, with application to fields with and without gaps in their flux surfaces. Gaps automatically produce tangential discontinuities in almost all cases, by permitting fields otherwise separated by finite distance to come in contact. Both Euclidean and non Euclidean flux surfaces are examined, showing that positive curvature fosters the formation of gaps while negative curvature opposes it. The special conditions for producing single or double gaps are pointed out. The general conditions for producing gaps are so mild as to indicate the special character of the familiar continuous solutions to the force-free equilibrium equations, in which the maxima and minima of the field pressure are so arranged as to provide flux surfaces without gaps.