We simulate the multi-phase interstellar medium (ISM) randomly heated and stirred by supernovae (SNe), with gravity, differential rotation and other parameters of the solar neighbourhood. Here we describe in detail both numerical and physical aspects of the model, including injection of thermal and kinetic energy by SN explosions, radiative cooling, photoelectric heating and various transport processes. With a three-dimensional domain extending 1 × 1 kpc 2 horizontally and 2 kpc vertically (symmetric about the galactic mid-plane), the model routinely spans gas number densities 10 −5 -10 2 cm −3 , temperatures 10-10 8 K, local velocities up to 10 3 km s −1 (with Mach number up to 25). The working numerical resolution of 4 pc has been selected via simulations of a single expanding SN remnant, where we closely reproduce, at this resolution, analytical solutions for the adiabatic and snowplough regimes. The feedback of the the halo on the disc cannot be captured in our model where the domain only extends to the height of 1 kpc above the mid-plane. We argue that to reliably model the disc-halo connections would require extending the domain horizontally as well as vertically due to the increasing horizontal scale of the gas flows with height.The thermal structure of the modelled ISM is classified by inspection of the joint probability density of the gas number density and temperature. We confirm that most of the complexity can be captured in terms of just three phases, separated by temperature borderlines at about 10 3 K and 5 × 10 5 K. The probability distribution of gas density within each phase is approximately lognormal. We clarify the connection between the fractional volume of a phase and its various proxies, and derive an exact relation between the fractional volume and the filling factors defined in terms of the volume and probabilistic averages. These results are discussed in both observational and computational contexts. The correlation scale of the random flows is calculated from the velocity autocorrelation function; it is of order 100 pc and tends to grow with distance from the mid-plane. We use two distinct parameterizations of radiative cooling to show that the multi-phase structure of the gas is robust, as it does not depend significantly on this choice.

The origin and structure of the magnetic fields in the interstellar medium of spiral galaxies is investigated with 3D, non-ideal, compressible MHD simulations, including stratification in the galactic gravity field, differential rotation and radiative cooling. A rectangular domain, 1 × 1 × 2 kpc 3 in size, spans both sides of the galactic mid-plane. Supernova explosions drive transonic turbulence. A seed magnetic field grows exponentially to reach a statistically steady state within 1.6 Gyr. Following Germano (1992) we use volume averaging with a Gaussian kernel to separate magnetic field into a mean field and fluctuations. Such averaging does not satisfy all Reynolds rules, yet allows a formulation of mean-field theory. The mean field thus obtained varies in both space and time. Growth rates differ for the mean-field and fluctuating field and there is clear scale separation between the two elements, whose integral scales are about 0.7 kpc and 0.3 kpc, respectively.

Magnetic fields grow quickly even at early cosmological times, suggesting the action of a small-scale dynamo (SSD) in the interstellar medium of galaxies. Many studies have focused on idealized turbulent driving of the SSD. Here we simulate more realistic supernova-driven turbulence to determine whether it can drive an SSD. Magnetic field growth occurring in our models appears inconsistent with simple tangling of magnetic fields, but consistent with SSD action, reproducing and confirming models by Balsara et al. that did not include physical resistivity η. We vary η, as well as the numerical resolution and supernova rate, , to delineate the regime in which an SSD occurs. For a given we find convergence for SSD growth rate with resolution of a parsec. For , with the solar neighborhood rate, the critical resistivity below which an SSD occurs is , and this increases with the supernova rate. Across the modeled range of 0.5–4 pc resolution we find that for , the SSD saturates at about 5% of kinetic energy equipartition, independent of growth rate. In the range growth rate increases with . SSDs in the supernova-driven interstellar medium commonly exhibit erratic growth.

We apply correlation analysis to random fields in numerical simulations of the supernova-driven interstellar medium (ISM) with the magnetic field produced by dynamo action. We solve the magnetohydrodynamic (MHD) equations in a shearing Cartesian box representing a local region of the ISM, subject to thermal and kinetic energy injection by supernova explosions, and parameterized, optically thin radiative cooling. We consider the cold, warm, and hot phases of the ISM separately; the analysis mostly considers the warm gas, which occupies the bulk of the domain. Various physical variables have different correlation lengths in the warm phase: 40, 50, and 60 pc for the random magnetic field, density, and velocity, respectively, in the midplane. The correlation time of the random velocity is comparable to the eddy turnover time, about 10 year 7 , although it may be shorter in regions with a higher star formation rate. The random magnetic field is anisotropic, with the standard deviations of its components b b b x y z having approximate ratios 0.5 0.6 0.6 in the midplane. The anisotropy is attributed to the global velocity shear from galactic differential rotation and locally inhomogeneous outflow to the galactic halo. The correlation length of Faraday depth along the z axis, 120 pc, is greater than for electron density, 60 90 pc -, and the vertical magnetic field, 60 pc. Such comparisons may be sensitive to the orientation of the line of sight. Uncertainties of the structure functions of synchrotron intensity rapidly increase with the scale. This feature is hidden in a power spectrum analysis, which can undermine the usefulness of power spectra for detailed studies of interstellar turbulence.