The paper studies field-line dynamics of divergence-free surface currents used to represent stellarator coils. On smooth toroidal winding surfaces, generic currents satisfy a dichotomy: a nowhere-vanishing current has either dense trajectories or periodic trajectories with a common winding type, whereas a current with zeros contains both centre and saddle regions and has only finitely many nonperiodic connecting trajectories. For cylindrical coil surfaces, oppositely directed currents on the two boundary circles generically force centre and saddle regions; nearly all remaining trajectories are periodic. Imposing steady-state Maxwell and Ohmic assumptions makes the current both divergence- and curl-free, hence a harmonic Neumann field. On a cylinder this space is one-dimensional, the nonzero field has no zeros, and every trajectory is a closed poloidal orbit, excluding centre and saddle regions. Additional recurrence and Poincar'e--Bendixson arguments show that almost every orbit of a general divergence-free cylindrical current is periodic. These results explain observed current patterns and identify assumptions under which coil contours simplify.
- Proved a generic toroidal-current dichotomy between nowhere-vanishing dense or periodic dynamics and dynamics with centre and saddle regions.
- Classified exceptional toroidal trajectories as singularities or finitely many saddle connections.
- Showed that opposite boundary-current orientations on a cylindrical surface generically force centre and saddle regions.
- Derived harmonic Neumann currents from steady-state Maxwell and linear isotropic Ohmic assumptions.
- Proved that nonzero harmonic cylindrical currents consist entirely of closed, noncontractible poloidal field lines.