We present a semi-analytic model for the interstellar medium that considers local processes and structures of turbulent star-forming gas. A volume element of the interstellar medium is described as a multi-phase system, comprising a cold and a warm gas phase in effective (thermal plus turbulent) pressure equilibrium and a stellar component. The cooling instability of the warm gas feeds the cold phase, while various heating processes transfer cold gas to the warm phase. The cold phase consists of clumps embedded in diffuse warm gas, where only the molecular fraction of the cold gas may be converted into stars. The fraction of molecular gas is approximately calculated, using a Strömgren-like approach and the efficiency of star formation is determined by the state of the cold gas and the turbulent velocity dispersion on the clump length-scale. Gas can be heated by supernovae and ultraviolet emission of massive stars, according to the evolutionary stages of the stellar populations and the initial mass function. Since turbulence has a critical impact on the shape of the gaseous phases, on the production of molecular hydrogen and on the formation of stars, the consistent treatment of turbulent energy – the kinetic energy of unresolved motions – is an important new feature of our model. Besides turbulence production by supernovae and the cooling instability, we also take into account the forcing by large-scale motions.
We formulate a set of ordinary differential equations, which statistically describes star formation and the exchange between the different budgets of mass and energy in a region of the interstellar medium with given mean density, size, metallicity and external turbulence forcing. By exploring the behaviour of the solutions, we find equilibrium states, in which the star formation efficiencies are consistent with observations. Kennicutt–Schmidt-like relations naturally arise from the equilibrium solutions, while conventional star formation models in numerical simulations impose such relations with observed efficiency parameters as phenomenological calibrations.
Beyond the semi-analytic approach, a potential application is a complete sub-grid scale model of the unresolved multi-phase structure, star formation and turbulence in simulations of galaxies or in cosmological simulations. The formulation presented in this article combines various models focusing on particular processes and yet can be adopted to specific applications, depending on the range of resolved length-scales.