  We suggest that large HI holes observed in the interstellar medium (ISM) of
galaxies such as the Large Magellanic Cloud (LMC) and Holmberg II (Ho II, DDO
50, UGC 4305) can form as the combined result of turbulence coupled to thermal
and gravitational instabilities. We investigate this problem with three
dimensional hydrodynamical simulations, taking into account cooling and heating
processes and the action of the self-gravity of the gas. We construct an
algorithm for radiative transfer to post-process the simulated data and build
emission maps in the 21 cm neutral hydrogen line. With this approach, we are
able to reproduce the structure of the shells and holes as observed in regions
of the ISM, where no stellar activity is detected. In order to quantify the
comparison of our synthetic maps to the observations, we calculate the physical
scale-autocorrelation length relation (L-L_{cr} relation) both on the synthetic
HI maps and the HI map of Ho II. The L-L_{cr} relation shows a linear increase
of the autocorrelation length with the physical scale up to the scale of energy
injection and flattens for larger scales. The comparison of the L-L_{cr}
relation between the observations and the synthetic maps suggests that
turbulence is driven in the ISM of Ho II on large scales (~ 6 kpc). The slope
of the L-L_{cr} relation in the linear regime in Ho II is better reproduced by
models where turbulence is coupled with a weak efficiency cooling of the gas.
These results demonstrate the importance of the interplay between turbulence
and the thermodynamics of the gas for structure formation in the ISM. Our
analysis can be used to determine the scale on which kinetic energy is injected
in the ISM of dwarf irregular galaxies, and to derive, in a first
approximation, the cooling rate of the gas.
