  We explore the emergence of universal dynamic scaling in an interacting Bose
gas around the condensation transition, under the combined influence of an
external driving force and spatial disorder. As time progresses, we find that
the Bose gas crosses over three distinct dynamical regimes: (i) an inverse
turbulent cascade where interactions dominate the drive, (ii) a stationary
regime where the inverse cascade and the drive counterbalance one other, and
(iii) a sub-diffusive cascade in energy space governed by the drive and
disorder, a phenomenon recently observed experimentally. We show that all three
dynamical regimes can be described by self-similar scaling laws.
