  Simulations generally show that non-self-gravitating clouds have a lognormal
column density ($\Sigma$) probability distribution function (PDF), while
self-gravitating clouds with active star formation develop a distinct power-law
tail at high column density. Although the growth of the power law can be
attributed to gravitational contraction leading to the formation of condensed
cores, it is often debated if an observed lognormal shape is a direct
consequence of supersonic turbulence alone, or even if it is really observed in
molecular clouds. In this paper we run three-dimensional magnetohydrodynamic
simulations including ambipolar diffusion with different initial conditions to
see the effect of strong magnetic fields and nonlinear initial velocity
perturbations on the evolution of the column density PDFs. Our simulations show
that column density PDFs of clouds with supercritical mass-to-flux ratio, with
either linear perturbations or nonlinear turbulence, quickly develop a
power-law tail such that $dN/d \log \Sigma \propto \Sigma^{-\alpha}$ with index
$\alpha \simeq 2$. Interestingly, clouds with subcritical mass-to-flux ratio
also proceed directly to a power-law PDF, but with a much steeper index $\alpha
\simeq 4$. This is a result of gravitationally-driven ambipolar diffusion.
However, for nonlinear perturbations with a turbulent spectrum ($v_{k}^{2}
\propto k^{-4}$), the column density PDFs of subcritical clouds do retain a
lognormal shape for a major part of the cloud evolution, and only develop a
distinct power-law tail with index $\alpha \simeq 2$ at greater column density
when supercritical pockets are formed.
