  Optomechanical pattern forming instabilities in a cloud of cold atoms lead to
self-organized spatial structures of light and atoms. Here, we consider the
optomechanical self-structuring of a cold atomic cloud in the presence of a
phase structured input field, carrying orbital angular momentum. For a planar
ring cavity setup, a model of coupled cavity field and atomic density equations
describes a wide range of drifting modulation instabilities in the transverse
plane. This leads to the formation of rotating self-organized rings of
light-atom lattices. Using linear stability analysis and numerical simulations
of the coupled atomic and optical dynamics, we demonstrate the presence of
macroscopic atomic transport corresponding to the pattern rotation, induced by
the structured pump phase profile
