Clebsch potentials and the visualization of three-dimensional solenoidal vector fields

Clebsch potentials are of tremendous value in visualizing the flux lines of a three-dimensional solenoidal vector field. Unfortunately, only a local theory of Clebsch potentials exists, and their use in visualizing arbitrary three-dimensional solenoidal vector fields requires a global theory. Furthermore, such a global theory must be built on constructive techniques which yield practical algorithms. The virtues of Clebsch potentials are reviewed and a question about their global existence is formulated. The subtlety of this question is illustrated through an analogy with micromagnetic theory. A formalism for addressing questions about Clebsch potentials is presented.