We have modelled the high-energy astrophysics of the inner 200 pc of the Galaxy with a view to explaining the diffuse, broad-band (radio continuum to TeV γ-ray), non-thermal signal detected from this region. Our modelling pins down the ISM parameters for the environment wherein cosmic ray (CR) electrons and ions reside in the Galactic centre (GC). We find that the magnetic field in this region is 100-300 μG, the gas density ~<60 cm^-3, and that a powerful (> 200 km/s) ‘super’-wind acts to remove > 95% of the cosmic rays accelerated in the region before they have time to lose their energy in situ. The ~10^39 erg/s carried away by the GC cosmic ray protons is precisely enough to energise the ~GeV γ-ray emission from the Fermi ‘bubbles’ recent...