  The formation of a thin current sheet in a magnetic quasi-separatrix layer
(QSL) is investigated by means of numerical simulation using a simplified
ideal, low-$\beta$, MHD model. The initial configuration and driving boundary
conditions are relevant to phenomena observed in the solar corona and were
studied earlier by Aulanier et al., A&A 444, 961 (2005). In extension to that
work, we use the technique of adaptive mesh refinement (AMR) to significantly
enhance the local spatial resolution of the current sheet during its formation,
which enables us to follow the evolution into a later stage. Our simulations
are in good agreement with the results of Aulanier et al. up to the calculated
time in that work. In a later phase, we observe a basically unarrested collapse
of the sheet to length scales that are more than one order of magnitude smaller
than those reported earlier. The current density attains correspondingly larger
maximum values within the sheet. During this thinning process, which is finally
limited by lack of resolution even in the AMR studies, the current sheet moves
upward, following a global expansion of the magnetic structure during the
quasi-static evolution. The sheet is locally one-dimensional and the plasma
flow in its vicinity, when transformed into a co-moving frame, qualitatively
resembles a stagnation point flow. In conclusion, our simulations support the
idea that extremely high current densities are generated in the vicinities of
QSLs as a response to external perturbations, with no sign of saturation.
