Seeds of Life in Space (SOLIS). III. Zooming into the methanol peak of
  the pre-stellar core L1544
  Towards the pre-stellar core L1544, the methanol (CH$_3$OH) emission forms an
asymmetric ring around the core centre, where CH$_3$OH is mostly in solid form,
with a clear peak 4000~au to the north-east of the dust continuum peak. As part
of the NOEMA Large Project SOLIS (Seeds of Life in Space), the CH$_3$OH peak
has been spatially resolved to study its kinematics and physical structure and
to investigate the cause behind the local enhancement. We find that methanol
emission is distributed in a ridge parallel to the main axis of the dense core.
The centroid velocity increases by about 0.2~km~s$^{-1}$ and the velocity
dispersion increases from subsonic to transonic towards the central zone of the
core, where the velocity field also shows complex structure. This could be
indication of gentle accretion of material onto the core or interaction of two
filaments, producing a slow shock. We measure the rotational temperature and
show that methanol is in local thermodynamic equilibrium (LTE) only close to
the dust peak, where it is significantly depleted. The CH$_3$OH column density,
$N_{tot}({\rm CH_3OH})$, profile has been derived with non-LTE radiative
transfer modelling and compared with chemical models of a static core. The
measured $N_{tot}({\rm CH_3OH})$ profile is consistent with model predictions,
but the total column densities are one order of magnitude lower than those
predicted by models, suggesting that the efficiency of reactive desorption or
atomic hydrogen tunnelling adopted in the model may be overestimated; or that
an evolutionary model is needed to better reproduce methanol abundance.
