  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 $H_{ext}$, in which a perforating (rim crossing) avalanche appears, is
calculated. A staircase dependence of the mean field trapped inside a ring,
$\left\langle H\right\rangle$, vs. $H_{ext}$, is comprehensively described. A
staircase slope appears to be a universal function of a ring shape (rim width
to ring 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. Our theoretical results are in a
good accordance with the experimental data.

Superconductivity