High-Kappa Limits of the Time-Dependent Ginzburg–Landau Model

Abstract

The time-dependent Ginzburg–Landau model of superconductivity is examined in the high-$\kappa $, high magnetic field setting. This work generalizes the previous result for the steady-state model with a constant applied magnetic field. The significance of this generalization lies in the ability to incorporate the effects of both the applied magnetic field and applied current or voltage. Thus, it is possible to use the simplified setting obtained in this paper to study the motion and “pinning” of vortices in the presence of an applied current and a variable applied field. The results within are obtained via a formal asymptotic expansion of the Ginzburg–Landau equations in terms of $\kappa$, which yields a simplified system of leading-order equations. The formal asymptotic expansion is then justified by showing the solution to the full time-dependent Ginzburg-Landau equations converges to the solution of the leading-order equations as $\kappa \to \infty $. Computational results are also given that show the simplified leading-order model is indeed an accurate approximation to the solution of the full system of equations, even for moderate values of $\kappa$.