  Gamma-ray data from Fermi-LAT show a bi-lobular structure extending up to 50
degrees above and below the Galactic centre, coincident with a possibly related
structure in the ROSAT X-ray map which presumably originated in some energy
release close to the centre a few million years ago. It has been argued that
the gamma-rays arise due to inverse Compton scattering of relativistic
electrons accelerated at plasma shocks present in the bubbles. We explore the
alternative possibility that the relativistic electrons undergo stochastic
2nd-order Fermi acceleration in the entire volume of the bubbles by plasma wave
turbulence. This turbulence is generated behind the outer shock and propagates
into the bubble volume, leading to a non-trivial spatial variation of the
electron spectral index. Rather than a constant volume emissivity as predicted
in other models we find an almost constant surface brightness in gamma-rays and
also reproduce the observed sharp edges of the bubbles. We comment on possible
cross-checks in other channels.
