Here we present our two-dimensional chemodynamical code CoDEx, which we developed for the purpose of modeling the evolution of galaxies in a self-consistent manner. The code solves the hydrodynamical and momentum equations for three stellar components and the multiphase interstellar medium (clouds and intercloud medium), including star formation, Type I and Type II supernovae, planetary nebulae, stellar winds, evaporation and condensation, drag, cloud collisions, heating and cooling, and stellar nucleosynthesis. These processes are treated simultaneously, coupling a large range in temporal and spatial scales, to account for feedback and self-regulation processes, which play an extraordinarily important role in the galactic evolution. The evolution of galaxies of different masses and angular momenta is followed through all stages from the initial protogalactic clouds until now. In this first paper we present a representative model of the Milky Way and compare it with observations. The capability of chemodynamical models is convincingly proved by the excellent agreement with various observations. In addition, well-known problems (the G-dwarf problem, the discrepancy between local effective yields, etc.), which so far could be only explained by artificial constraints, are also solved in the global scenario. Starting from a rotating protogalactic gas cloud in virial equilibrium, which collapses owing to dissipative cloud-cloud collisions, we can follow the galactic evolution in detail. Owing to the collapse, the gas density increases, stars are forming, and the first Type II supernovae explode. The collapse time is 1 order of magnitude longer than the dynamical free-fall time because of the energy release by Type II supernovae. The supernovae also drive hot metal-rich gas ejected from massive stars into the halo, and as a consequence, the clouds in the star-forming regions have lower metallicities than the clouds in the halo. The observed negative metallicity gradients do not form before t = 6 × 109 yr. These outward gas flows prevent any clear correlation between local star formation rate and enrichment and also prevent a unique age-metallicity relation. The situation, however, is even more complicated, because the mass return of intermediate-mass stars (Type I supernovae and planetary nebulae) is delayed depending on the type of precursor. Since our chemodynamical model includes all these processes, we can calculate, e.g., the [O/H] distribution of stars and find good agreement everywhere in bulge, disk, and halo. From the galactic oxygen to iron ratio, we can determine the supernovae ([II + Ib]/Ia) ratio for different types of Type Ia supernovae (such as carbon deflagration or sub-Chandrasekhar models) and find that the ratio should be in the range 1.0-3.8. The chemodynamical model also traces other chemical elements (e.g., N + C), density distributions, gas flows, velocity dispersions of the stars and clouds, star formation, planetary nebula rates, cloud collision, condensation and evaporation rates, and the cooling due to radiation. The chemodynamical treatment of galaxy evolution should be envisaged as a necessary development, which takes those processes into account that affect the dynamical, energetical, and chemical evolution.
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