  We study the variational convergence of a family of two-dimensional
Ginzburg-Landau functionals arising in the study of superfluidity or thin-film
superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this
regime and for large enough applied rotations (for superfluids) or magnetic
fields (for superconductors), the minimizers acquire quantized point
singularities (vortices). We focus on situations in which an unbounded number
of vortices accumulate along a prescribed Jordan curve or a simple arc in the
domain. This is known to occur in a circular annulus under uniform rotation, or
in a simply connected domain with an appropriately chosen rotational vector
field. We prove that, suitably normalized, the energy functionals
Gamma-converge to a classical energy from potential theory. Applied to global
minimizers, our results describe the limiting distribution of vortices along
the curve in terms of Green equilibrium measures.
