Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity

This paper continues our study of vortices in Ginzburg-Landau theories with special attention to applications in superconductivity. In another paper, we derived asymptotic equations governing the dynamics of interacting vortices. Here, we study the hydrodynamic limit of these vortices. For vortices in the solutions of the nonlinear Schrödinger equation, the hydrodynamic equation is the incompressible Euler's equation in fluid mechanics. For vortices in the time-dependent Ginzburg-Landau equations, the hydrodynamic equations can be thought of as being the complement of the Euler equations. Preliminary results on the numerical studies of the hydrodynamic equations are presented. As applications of the hydrodynamic formalism, we study the pinning of vortex liquids by periodic potentials, and the propagation of magnetic fields into type-II superconductors. The hydrodynamic formalism suggests that to leading order, the vortex liquids are pinned even at small but positive temperature.