  We perform a long and accurate Large-Eddy Simulation of a binary neutron star
merger, following the newly formed remnant up to 110 milliseconds. The
combination of high-order schemes, high-resolution and the gradient
subgrid-scale model allow us to have among the highest effective resolutions
ever achieved. Our results show that, although the magnetic fields are strongly
amplified by the Kelvin-Helmholtz instability, they are coherent only over very
short spatial scales until t \gtrsim 30 ms. Around that time, magnetic winding
becomes more efficient leading to a linear growth of the toroidal component and
slowly ordering the field to more axisymmetric, large scales. The poloidal
component only starts to grow at small scales at much later times t \gtrsim 90
ms, in a way compatible with the magneto-rotational instability. No strong
large-scale poloidal field or jet is produced in the timescales spanned by our
simulation, although there is an helicoidal structure gradually developing at
late times. We highlight that soon after the merger the topology is always
strongly dominated by toroidal structures, with a complex distribution in the
meridional plane and highly turbulent perturbations. Thus, starting with strong
purely dipolar fields before the merger is largely inconsistent with the
outcomes of a realistic evolution. Finally, we confirm the universality of the
evolved topology, even when starting with very different magnetic fields
confined to the outermost layers of each neutron star.
