ISLANDS IN 3-DIMENSIONAL STEADY FLOWS

We consider the problem of steady Euler flows in a torus. We show that in the absence of a direction of symmetry the solution for the vorticity contains delta-function singularities at the rational surfaces of the torus. We study the effect of a small but finite viscosity on these singularities. The solutions near a rational surface contain cat's eyes islands, well known in the classical theory of critical layers. When the islands are small, their widths can be computed by a boundary-layer analysis. We show that the islands at neighbouring rational surfaces generally overlap. Thus, steady toroidal flows exhibit a tendency towards Beltramization.