Volume filling factors of the ISM phases in star forming galaxies - I. The role of the disk-halo interaction

The role of matter circulation between the disk and halo in establishing the volume filling factors of the different ISM phases in the Galactic disk (\z\less than or equal to250 pc) is investigated, using a modified version of the three-dimensional supernova-driven ISM model of Avillez (2000). We carried out adaptive mesh refinement simulations of the ISM with five supernova rates (in units of the Galactic value), sigma/sigma(Gal)=1, 2, 4, 8 and 16 (corresponding to starburst conditions) using three finer level resolutions of 2.5, 1.25 and 0.625 pc, allowing us to understand how resolution would affect the volumes of gas phases in pressure equilibrium. We find that the volume filling factors of the different ISM phases depend sensitively on the existence of a duty cycle between the disk and halo acting as a pressure release valve for the hot (T>10(5.5) K) phase in the disk. The amount of cold gas (defined as the gas with T<10(3) K) picked up in the simulations varies from a value of 19% for sigma/sigma(Gal)=1 to similar to 5% for sigma/sigma(Gal)=4 and <= 1% for higher SN rates. Background heating prevents the cold gas from immediate collapse and thus ensures the stability of the cold gas phase. The mean occupation fraction of the hot phase varies from about 17% for the Galactic SN rate to similar to 28%, for sigma/sigma(Gal)=4, and to 44% for sigma/sigma(Gal)=16. Overall the filling factor of the hot gas does not increase much as we move towards higher SN rates, following a power law of < f(v,hot)> proportional to (sigma/sigma(Gal))(0.363). Such a modest dependence on the SN rate is a consequence of the evacuation of the hot phase into the halo through the duty cycle. This leads to volume filling factors of the hot phase considerably smaller than those predicted in the three-phase model of McKee & Ostriker (1977) even in the absence of magnetic fields.