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We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg–Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the \({\Gamma}\)-limit of a sequence of scaled Ginzburg–Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg–Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg–Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg–Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations