We present a model for star formation and supernova feedback that describes the multi-phase structure of star forming gas on scales that are typically not resolved in cosmological simulations. Our approach includes radiative heating and cooling, the growth of cold clouds embedded in an ambient hot medium, star formation in these clouds, feedback from supernovae in the form of thermal heating and cloud evaporation, galactic winds and outflows, and metal enrichment. Implemented using SPH, our scheme is a significantly modified and extended version of the grid-based method of Yepes et al. (1997), and enables us to achieve high dynamic range in simulations of structure formation. We discuss properties of the feedback model in detail and show that it predicts a self-regulated, quiescent mode of star formation, which, in particular, stabilises the star forming gaseous layers of disk galaxies. The parameterisation of this mode can be reduced to a single free quantity which determines the overall timescale for star formation. We fix this parameter to match the observed rates of star formation in local disk galaxies. When normalised in this manner, cosmological simulations nevertheless overproduce the observed cosmic abundance of stellar material. We are thus motivated to extend our feedback model to include galactic winds associated with star formation. Using small-scale simulations of individual star-forming disk galaxies, we show that these winds produce either galactic fountains or outflows, depending on the depth of the gravitational potential. Moreover, outflows from galaxies in these simulations drive chemical enrichment of the intergalactic medium, in principle accounting for the presence of metals in the Lyman alpha forest. (abridged)
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