  We study the dynamics of a black hole in an asymptotically $AdS_d \times S^d$
space-time in the limit of a large number of dimensions, $d \to \infty$. Such a
black hole is known to become dynamically unstable below a critical radius. We
derive the dispersion relation for the quasinormal mode that governs this
instability in an expansion in $1/d$. We also provide a full nonlinear analysis
of the instability at leading order in $1/d$. We find solutions that resemble
the lumpy black spots and black belts previously constructed numerically for
small $d$, breaking the $SO(d+1)$ rotational symmetry of the sphere down to
$SO(d)$. We are also able to follow the time evolution of the instability. Due
possibly to limitations in our analysis, our time dependent simulations do not
settle down to stationary solutions. This work has relevance for strongly
interacting gauge theories; through the AdS/CFT correspondence, the special
case $d=5$ corresponds to maximally supersymmetric Yang-Mills theory on a
spatial $S^3$ in the microcanonical ensemble and in a strong coupling and large
number of colors limit.
