Where infall meets outflows: turbulent dissipation probed by CH${}^{+}$ and Ly$\alpha$ in the starburst/AGN galaxy group SMM J02399$-$0136 at z$\sim$2.8

We present a comparative analysis of the $\mathrm{C}{\mathrm{H}}^{+}$(1-0) and $\mathrm{L}\mathrm{y}\alpha$ lines, observed with the Atacama Large Millimeter Array (ALMA) and Keck telescope respectively, in the field of the submillimetre-selected galaxy (SMG) SMM,J02399$-$0136 at $z\sim 2.8$, which comprises a heavily obscured starburst galaxy and a broad absorption line quasar, immersed in a large $\mathrm{L}\mathrm{y}\alpha$ nebula. This comparison highlights the critical role played by turbulence in channeling the energy across gas phases and scales, splitting the energy trail between hot/thermal and cool/turbulent phases in the circum-galactic medium (CGM). The unique chemical and spectroscopic properties of $\mathrm{C}{\mathrm{H}}^{+}$ are used to infer the existence of a massive ($\sim 3.5\times {10}^{10}$ ${\rm M}_\odot$),highlyturbulentreservoirofdiffusemoleculargasofradius$\sim 20,$kpccoincidingwiththecoreofthe$\rm Ly \alpha$ nebula. The whole cool and cold CGM is shown to be inflowing towards the galaxies at a velocity $\sim$ 400 km$s^{-1}$. Several kpc-scale shocks are detected tentatively in $\rm CH^+$ emission. Their specific location in space and velocity with respect to the high-velocity $\mathrm{L}\mathrm{y}\alpha$ emission suggests that they lie at the interface of the inflowing CGM and the high-velocity $\mathrm{L}\mathrm{y}\alpha$ emission, and signpost the feeding of CGM turbulence by AGN- and stellar-driven outflows. The mass and energy budgets of the CGM require net mass accretion at a rate commensurate with the star formation rate (SFR). From this similarity, we infer that the merger-driven burst of star formation and black-hole growth are ultimately fuelled by large-scale gas accretion.