  We consider a cosmological model with an interaction between dark matter and
dark energy which leaves the background cosmology unaffected and only affects
the evolution of the perturbations. This is achieved by introducing a coupling
given in terms of the relative velocities of dark matter and dark energy. This
interaction has the distinctive feature of appearing predominantly on small
scales, where peculiar velocities can become important. We confront the
predictions of the model to cosmological observations and find a potential
alleviation of the known tension in the amplitude of density perturbations as
measured by low redshift galaxy surveys and the Planck data. The model also
predicts a shift in the turnover of the matter power spectrum which does not
depend on the horizon at equality (fixed by the background cosmology and, thus,
unaffected by the perturbations) and is entirely due to the interaction between
dark matter and dark energy. A bias in the peculiar velocity between baryons
and dark matter is also shown to be a unique feature of this type of
interactions in the dark sector.
