**Abstract** : We study the gluon cascade generated via successive medium-induced branchings by an energetic parton propagating through a dense QCD medium. We focus on the high-energy regime where the energy $E$ of the leading particle is much larger than the characteristic medium scale $\omega_c=\hat q L^2/2$, with $\hat q$ the jet quenching parameter and $L$ the distance travelled through the medium. In this regime the leading particle loses only a small fraction $\sim\alpha_s(\omega_c/E)$ of its energy and can be treated as a steady source of radiation for gluons with energies $\omega\le\omega_c$. For this effective problem with a source, we obtain exact analytic solutions for the gluon spectrum and the energy flux. The solutions exhibit wave turbulence: the basic physical process is a continuing fragmentation which is `quasi-democratic' (i.e. quasi-local in energy) and which provides an energy transfer from the source to the medium at a rate (the energy flux $\mathcal{F}$) which is quasi-independent of $\omega$. The locality of the branching process implies a spectrum of the Kolmogorov-Obukhov type, i.e. a power-law spectrum which is a fixed point of the branching process and whose strength is proportional to the energy flux: $D(\omega)\sim\mathcal{F}/\sqrt\omega$ for $\omega\ll\omega_c$. Via this turbulent flow, the gluon cascade loses towards the medium an energy $\Delta E\sim\alpha_s^2\omega_c$, which is independent of the initial energy $E$ of the leading particle and of the details of the thermalization mechanism at the low-energy end of the cascade. This energy is carried away by very soft gluons, which propagate at very large angles with respect to the jet axis. Our predictions for the value of $\Delta E$ and for its angular distribution appear to agree quite well, qualitatively and even semi-quantitatively, with the phenomenology of di-jet asymmetry in nucleus-nucleus collisions at the LHC.