Avalanches in the Bean critical-state model

A macroscopic equation of motion for the flux density in dirty type-ii superconductors is introduced. The flux density is subject to various types of spatially varying pinning force. When there is-no stick-slip dynamics, i.e., when the static pinning force equals the dynamic pinning force, it is shown that in both one and two dimensions an increase in the surface magnetic field leads to an overall height change and hence to a change in magnetization equal to the change in the surface magnetic field. More interesting behavior occurs on introducing stick-slip dynamics, i.e., when the static pinning force exceeds the dynamic pinning force. In this limit a distribution of avalanche sizes over four orders of magnitude is found for a 100 x 100 lattice. Apart from the anomalous behavior at large sizes, this is shown to fit a distribution of the form P(s)approximate to s(-nu) exp(-s/alpha)), where s is the avalanche size. The anomalous behavior for large sizes corresponds to avalanches which involve most of the lattice and, hence, cause the flux to ''slide over the edge,'' as detected by a change in the edge magnetization.