  In this paper we focus on understanding the physical processes that lead to
stable or unstable ionization fronts (I-fronts) observed in simulations of
moving black holes (BHs). The front instability may trigger bursts of gas
accretion, rendering the BH significantly more luminous than at steady-state.
We perform a series of idealized three dimensional radiation hydrodynamics
simulations resolving the I-fronts around BHs of mass $M_\mathrm{BH}$ and
velocity $v_\infty$ accreting from a medium of density $n_\mathrm{H}$. The
I-front, with radius $R_\mathrm{I}$, transitions from D-type to R-type as the
BH velocity becomes larger than a critical value $v_\mathrm{R}\sim
40\,\mathrm{km/s}$. The D-type front is preceded by a bow-shock of thickness
$\Delta R_\mathrm{I}$ that decreases as $v_\infty$ approaches $v_\mathrm{R}$.
We find that both D-type and R-type fronts can be unstable given the following
two conditions: i) for D-type fronts the shell thickness must be $\Delta
R_\mathrm{I}/R_\mathrm{I}<0.05$ (i.e., $v_\infty \gtrsim 20\,\mathrm{km/s}$.),
while no similar restriction holds for R-type fronts; ii) the temperature jump
across the I-front must be $T_\mathrm{II}/T_\mathrm{I}>3$. This second
condition is satisfied if $T_\mathrm{I}<5000\,\mathrm{K}$ or if
$n_\mathrm{H}\,M_\mathrm{BH} \gtrsim 10^6\,M_\odot\,\mathrm{cm^{-3}}$. Due to
X-ray pre-heating typically $T_\mathrm{I} \sim 10^4\,\mathrm{K}$, unless the
D-type shell is optically thick to X-rays, which also happens when
$n_\mathrm{H}\,M_\mathrm{BH}$ is greater than a metallicity-dependent critical
value. We thus conclude that I-fronts around BHs are unstable only for
relatively massive BHs moving trough very dense molecular clouds. We briefly
discuss the observational consequences of the X-ray luminosity bursts likely
associated with this instability.
