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We present a comparative analysis of the $\rm CH^+$(1-0) and $\rm Ly \alpha$lines, observed with the Atacama Large Millimeter Array (ALMA) and Kecktelescope respectively, in the field of the submillimetre-selected galaxy (SMG)SMM\,J02399$-$0136 at $z\sim2.8$, which comprises a heavily obscured starburstgalaxy and a broad absorption line quasar, immersed in a large $\rm Ly \alpha$nebula. This comparison highlights the critical role played by turbulence inchanneling the energy across gas phases and scales, splitting the energy trailbetween hot/thermal and cool/turbulent phases in the circum-galactic medium(CGM). The unique chemical and spectroscopic properties of $\rm CH^+$ are usedto infer the existence of a massive ($\sim 3.5 \times 10^{10}$ ${\rmM}_\odot$), highly turbulent reservoir of diffuse molecular gas of radius $\sim20\,$kpc coinciding with the core of the $\rm Ly \alpha$ nebula. The whole cooland 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 $\rmCH^+$ emission. Their specific location in space and velocity with respect tothe high-velocity $\rm Ly \alpha$ emission suggests that they lie at theinterface of the inflowing CGM and the high-velocity $\rm Ly \alpha$ 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 ratecommensurate with the star formation rate (SFR). From this similarity, we inferthat the merger-driven burst of star formation and black-hole growth areultimately fuelled by large-scale gas accretion.

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