  We present numerical experiments aimed to study the correlation between the
magnetic field strength, $B$, and the density, $n$, in the cold atomic
interstellar medium (CNM). We analyze 24 magneto-hydrodynamic models with
different initial magnetic field intensities ($B_0=$0.4, 2.1, 4.2, and 8.3
$\mu$G) and/or mean densities (2, 3, and 4 cm$^{-3}$), in the presence of
driven and decaying turbulence, with and without self-gravity, in a cubic
computational domain with 100 pc by side. Our main findings are: i) For forced
simulations, which reproduce the main observed physical conditions of the CNM
in the Solar neighborhood, a positive correlation between $B$ and $n$ develops
for all the $B_0$ values. ii) The density at which this correlation becomes
significant ($\lesssim 30$ cm$^{-3}$) depends on $B_0$ but is not sensitive to
the presence of self-gravity. iii) The effect of self-gravity, when noticeable,
consists of producing a shallower correlation at high densities, suggesting
that, in the studied regime, self-gravity induces motions along the field
lines. iv) Self-gravitating decaying models where the CNM is subsonic and
sub-Alfv\'enic with $\beta\lesssim 1$ develop a high density positive
correlation whose slopes are consistent with a constant $\beta(n)$. v) Decaying
models where the low density CNM is subsonic and sub-Alfv\'enic with $\beta>1$
show a negative correlation at intermediate densities, followed by a high
density positive correlation.
