  Understanding star formation is problematic as it originates in the large
scale dynamics of a galaxy but occurs on the small scale of an individual star
forming event. This paper presents the first numerical simulations to resolve
the star formation process on sub-parsec scales, whilst also following the
dynamics of the interstellar medium (ISM) on galactic scales. In these models,
the warm low density ISM gas flows into the spiral arms where orbit crowding
produces the shock formation of dense clouds, held together temporarily by
their external pressure. Cooling allows the gas to be compressed to
sufficiently high densities that local regions collapse under their own gravity
and form stars. The star formation rates follow a Schmidt-Kennicutt
\Sigma_{SFR} ~ \Sigma_{gas}^{1.4} type relation with the local surface density
of gas while following a linear relation with the cold and dense gas. Cooling
is the primary driver of star formation and the star formation rates as it
determines the amount of cold gas available for gravitational collapse. The
star formation rates found in the simulations are offset to higher values
relative to the extragalactic values, implying a constant reduction, such as
from feedback or magnetic fields, is likely to be required. Intriguingly, it
appears that a spiral or other convergent shock and the accompanying thermal
instability can explain how star formation is triggered, generate the physical
conditions of molecular clouds and explain why star formation rates are tightly
correlated to the gas properties of galaxies.
