We present the results of particle level simulations of the propagation of cosmic ray air shower cores through high-altitude polar ice sheets, using both the COSIKA Monte Carlo code and the Geant4 simulation toolkit. We discuss the general features of the in-ice particle cascade, covering the deposited energy density, the longitudinal development of the shower, and a general parameterization of the radial charge distribution of the cascade front in function of Xmax. We present preliminary calculations of the Askaryan radio emission of the in-ice particle cascades. We find that the core of the air shower dominates the emission during the propagation in the ice, which may mimic an in-ice neutrino-induced particle cascade. Finally, we discuss the feasibility of using RADAR echo techniques to detect the plasma created in the ice by the particle cascade.