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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.

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The propagation of charged particles, including cosmic rays, in a partially ordered magnetic field is characterized by a diffusion tensor whose components depend on the particle's Larmor radius R L and the degree of order in the magnetic field. Most studies of the particle diffusion presuppose a scale separation between the mean and random magnetic fields (e.g., there being a pronounced minimum in the magnetic power spectrum at intermediate scales). Scale separation is often a good approximation in laboratory plasmas, but not in most astrophysical environments such as the interstellar medium (ISM). Modern simulations of the ISM have numerical resolution of order 1 pc, so the Larmor radius of the cosmic rays that dominate in energy density is at least 10 6 times smaller than the resolved scales. Large-scale simulations of cosmic ray propagation in the ISM thus rely on oversimplified forms of the diffusion tensor. We take the first steps towards a more realistic description of cosmic ray diffusion for such simulations, obtaining direct estimates of the diffusion tensor from test particle simulations in random magnetic fields (with the Larmor radius scale being fully resolved), for a range of particle energies corresponding to 10 −2 R L /l c 10 3 , where l c is the magnetic correlation length. We obtain explicit expressions for the cosmic ray diffusion tensor for R L /l c ≪ 1, that might be used in a sub-grid model of cosmic ray diffusion. The diffusion coefficients obtained are closely connected with existing transport theories that include the random walk of magnetic lines.