Staircase penetration of magnetic flux into a superconducting flat ring

We analyzed the distribution of the Meissner shielding currents in a flat superconducting ring and quantitatively described the penetration of magnetic avalanches (dendrites) inside it. Using a recurrent procedure, the external field Hext, in which a perforating (edge to edge crossing) avalanche appears, is calculated. A staircase dependence of the mean field trapped inside a ring, (H) vs Hext, is comprehensively described. A staircase slope appears to be a universal function of a ring shape (inner to outer diameter ratio). The heat released by a penetrating dendrite grows linearly with each next perforation. Flux pinning, if present, modifies a staircase dependence and makes steps smaller, but does not change the staircase slope. Our theoretical results are in a good accordance with the experimental data.