  The detection of binary neutron star mergers represents one of the most
important and complex astrophysical discoveries of the recent years. One of the
unclear aspects of the problem is the turbulent magnetic field amplification,
initially triggered by the Kelvin-Helmholtz instability at much smaller scales
than any reachable numerical resolution nowadays. Here we present numerical
simulations of the first ten milliseconds of a binary neutron star merger.
First, we confirm in detail how the simulated amplification depends on the
numerical resolution and is distributed on a broad range of scales, as expected
from turbulent MHD theory. We find that an initial large-scale magnetic field
of $10^{11}\,$G inside each star is amplified in the remnant to
root-mean-square values above $10^{16}\,$G within the first $5$ milliseconds
for our highest-resolution run. Then, we run large eddy simulations, exploring
the performance of the subgrid-scale gradient model, already tested
successfully in previous turbulent box simulations. We show that the addition
of this model is especially important in the induction equation, since it leads
to an amplification of the magnetic field comparable to a higher-resolution
run, but with a greatly reduced computational cost. In the first 10
milliseconds, there is no clear hint for an ordered, large-scale magnetic
field, which should indeed occur in longer timescales through magnetic winding
and the magneto-rotational instability.
