Journal of Engineering Mathematics
, Volume 85, Issue 1, pp 149177
First online:
Mathematical modelling of the effect of surface roughness on magnetic field profiles in type II superconductors
 Michael LindstromAffiliated withMathematics Department, University of British Columbia Email author
 , Brian WettonAffiliated withMathematics Department, University of British Columbia
 , Rob KieflAffiliated withPhysics and Astronomy Department, University of British Columbia
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One of the defining characteristics of a superconductor is the Meissner effect, in which an external magnetic field is expelled from the bulk of a sample when cooled below the critical temperature. Although there has been considerable theoretical work on the Ginzburg–Landau theory of superconductors, the effects of interest in this paper can be modelled with the simpler London equation. This equation predicts an exponential decay of the local magnetic field magnitude as a function of the distance into the superconductor from a flat surface in the London limit where \(\kappa =\lambda /\xi \), defined as the ratio between the penetration depth and coherence length, is much greater than 1. However, recent measurements of the field profile in high \(\kappa \) superconductors show that the observed decay is nonexponential near the surface. In particular, the measured field profiles indicate that the decay rate in the field magnitude is smaller than expected from a simple London model on a short length scale \(d\) near the surface. In this paper, we examine the effects of surface roughness on magnetic field penetration into a high \(\kappa \) superconductor. We model the roughness as a sinusoidal perturbation from a flat interface and investigate the effect using both an asymptotic method, based upon a smallamplitude perturbation, and a numerical method, using a finite difference discretization with a coordinate mapping from an underlying rectangular domain. A novel discretization is used in the case of 3D calculations and a fast, preconditioned GMRES solver is developed. A careful comparison of asymptotic and numerical methods validates both approaches for small perturbations, but the numerical approach allows for the investigation of rougher surfaces. Our results show that surface roughness reduces the decay rate in the average magnetic field near the surface relative to a London model. However, the reduction is more gradual than the simple dead layer model currently being used to fit experimental data. In addition, we discover some interesting new phenomena in the 3D case.
Keywords
Asymptotics Finite differences London equation Superconductor Surface roughness Title
 Mathematical modelling of the effect of surface roughness on magnetic field profiles in type II superconductors
 Journal

Journal of Engineering Mathematics
Volume 85, Issue 1 , pp 149177
 Cover Date
 201404
 DOI
 10.1007/s106650139640y
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Asymptotics
 Finite differences
 London equation
 Superconductor
 Surface roughness
 Industry Sectors
 Authors

 Michael Lindstrom ^{(1)}
 Brian Wetton ^{(1)}
 Rob Kiefl ^{(2)}
 Author Affiliations

 1. Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
 2. Physics and Astronomy Department, University of British Columbia, 6224 Agricultural Rd, Vancouver, BC, V6T 1Z1, Canada