The concept of "ion atmosphere" is prevalent in both theoretical and experimental studies of nucleic acid systems, yet the spatial arrangement and the composition of ions in the ion atmosphere remain elusive, in particular when several ionic species (e.g., Na+, K+, and Mg2+) compete to neutralize the charge of a nucleic acid polyanion. Complementing the experimental study of Bai and co-workers (J. Am. Chem. Soc. 2007, 129, 14981), here we characterize ion atmosphere around double-stranded DNA through all-atom molecular dynamics simulations. We demonstrate that our improved parametrization of the all-atom model can quantitatively reproduce the experimental ion-count data. Our simulations determine the size of the ion atmosphere, the concentration profiles of ionic species competing to neutralize the DNA charge, and the sites of the cations' preferential binding at the surface of double-stranded DNA. We find that the effective size of the ion atmosphere depends on both the bulk concentration and valence of ions: increasing either reduces the size of the atmosphere. Near DNA, the concentration of Mg2+ is strongly enhanced compared to monovalent cations. Within the DNA grooves, the relative concentrations of cations depend on their bulk values. Nevertheless, the total charge of competing cations buried in the DNA grooves is constant and compensates for about similar to 30% of the total DNA charge.