To study how supernova feedback structures the turbulent interstellar medium, we construct 3D models of vertically stratified gas stirred by discrete supernova explosions, including vertical gravitational field and parametrized heating and cooling. The models reproduce many observed characteristics of the Galaxy such as global circulation of gas (i.e., galactic fountain) and the existence of cold dense clouds in the galactic disk. Global quantities of the model such as warm and hot gas filling factors in the midplane, mass fraction of thermally unstable gas, and the averaged vertical density profile are compared directly with existing observations, and shown to be broadly consistent. We find that energy injection occurs over a broad range of scales. There is no single effective driving scale, unlike the usual assumption for idealized models of incompressible turbulence. However, >90% of the total kinetic energy is contained in wavelengths shortward of 200 pc. The shape of the kinetic energy spectrum differs substantially from that of the velocity power spectrum, which implies that the velocity structure varies with the gas density. Velocity structure functions demonstrate that the phenomenological theory proposed by Boldyrev is applicable to the medium. We show that it can be misleading to predict physical properties such as the stellar initial mass function based on numerical simulations that do not include self-gravity of the gas. Even if all the gas in turbulently Jeans unstable regions in our simulation is assumed to collapse and form stars in local freefall times, the resulting total collapse rate is significantly lower than the value consistent with the input supernova rate. Supernova-driven turbulence inhibits star formation globally rather than triggering it.
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