  Electromagnetic field confinement due to plasma near accreting black holes
can trigger superradiant instabilities at the linear level, limiting the spin
of black holes and providing novel astrophysical sources of electromagnetic
bursts. However, nonlinear effects might jeopardize the efficiency of the
confinement, rending superradiance ineffective. Motivated by understanding
nonlinear interactions in this scenario, here we study the full $3+1$ nonlinear
dynamics of Maxwell equations in the presence of plasma by focusing on regimes
that are seldom explored in standard plasma-physics applications, namely a
generic electromagnetic wave of very large amplitude but small frequency
propagating in an inhomogeneous, overdense plasma. We show that the plasma
transparency effect predicted in certain specific scenarios is not the only
possible outcome in the nonlinear regime: plasma blow-out due to nonlinear
momentum transfer is generically present and allows for significant energy
leakage of electromagnetic fields above a certain threshold. We argue that such
effect is sufficient to dramatically quench the plasma-driven superradiant
instability around black holes even in the most optimistic scenarios.
