  Dense molecular filaments are central to the star formation process, but the
detailed manner in which they fragment into prestellar cores is not yet well
understood. Here, we investigate the fragmentation properties and dynamical
state of several star-forming filaments in the X-shape Nebula region of the
California MC, in an effort to shed some light on this issue. We used
multi-wavelength far-infrared images from Herschel and the getsources and
getfilaments extraction methods to identify dense cores and filaments and
derive their basic properties. We also used a map of $\rm ^{13}CO (2-1)$
emission from SMT 10m submillimeter telescope to constrain the dynamical state
of the filaments. We identified 10 filaments, as well as 57 dense cores. Two
star-forming filaments (# 8 and # 10) stand out in that they harbor
quasi-periodic chains of dense cores with a typical projected core spacing of
$\sim$0.15 pc. These two filaments have thermally supercritical line masses and
are not static. Filament~8 exhibits a prominent transverse velocity gradient,
suggesting that it is accreting gas from the parent cloud gas reservoir. In
both cases, the observed (projected) core spacing is similar to the filament
width and significantly shorter than the canonical separation of $\sim \,$4
times the filament width predicted by classical cylinder fragmentation theory.
We suggest that continuous accretion of gas onto the two star-forming
filaments, as well as geometrical bending of the filaments, may account for the
observed core spacing. Our findings suggest that the characteristic
fragmentation lengthscale of molecular filaments is quite sensitive to external
perturbations from the parent cloud, such as gravitational accretion of ambient
material.
