  We consider three-dimensional AdS gravity minimally coupled to a massless
scalar field and study numerically the evolution of small smooth circularly
symmetric perturbations of the $AdS_3$ spacetime. As in higher dimensions, for
a large class of perturbations, we observe a turbulent cascade of energy to
high frequencies which entails instability of $AdS_3$. However, in contrast to
higher dimensions, the cascade cannot be terminated by black hole formation
because small perturbations have energy below the black hole threshold. This
situation appears to be challenging for the cosmic censor. Analysing the energy
spectrum of the cascade we determine the width $\rho(t)$ of the analyticity
strip of solutions in the complex spatial plane and argue by extrapolation that
$\rho(t)$ does not vanish in finite time. This provides evidence that the
turbulence is too weak to produce a naked singularity and the solutions remain
globally regular in time, in accordance with the cosmic censorship hypothesis.
