  Dendritic flux avalanches is a frequently encountered consequence of the
thermomagnetic instability in type-II superconducting films. The avalanches,
potentially harmful for superconductor-based devices, can be suppressed by an
adjacent normal metal layer, even when the two layers are not in thermal
contact. The suppression of the avalanches in this case is due to so-called
magnetic braking, caused by eddy currents generated in the metal layer by
propagating magnetic flux. We develop a theory of magnetic braking by analyzing
coupled electrodynamics and heat flow in a superconductor-normal metal bilayer.
The equations are solved by linearization and by numerical simulation of the
avalanche dynamics. We find that in an uncoated superconductor, even a uniform
thermomagnetic instability can develop into a dendritic flux avalanche. The
mechanism is that a small non-uniformity caused by the electromagnetic
non-locality induces a flux-flow hot spot at a random position. The hot spot
quickly develops into a finger, which at high speeds penetrates into the
superconductor, forming a branching structure. Magnetic braking slows the
avalanches, and if the normal metal conductivity is sufficiently high, it can
suppress the formation of the dendritic structure. During avalanches, the
braking by the normal metal layer prevents the temperature from exceeding the
transition temperature of the superconductor. Analytical criteria for the
instability threshold are developed using the linear stability analysis. The
criteria are found to match quantitatively the instability onsets obtained in
simulation.
