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We analyzed the distribution of the Meissner shielding currents in a flatsuperconducting ring and quantitatively described the penetration of magneticavalanches (dendrites) inside it. Using a recurrent procedure, the externalfield $H_{ext}$, in which a perforating (rim crossing) avalanche appears, iscalculated. A staircase dependence of the mean field trapped inside a ring,$\left\langle H\right\rangle$, vs. $H_{ext}$, is comprehensively described. Astaircase slope appears to be a universal function of a ring shape (rim widthto ring diameter ratio). The heat released by a penetrating dendrite growslinearly with each next perforation. Flux pinning, if present, modifies astaircase dependence and makes steps smaller. Our theoretical results are in agood accordance with the experimental data.

Staircase penetration of magnetic flux into a superconducting flat ring