It has been suggested that galactic shock asymmetry induced by our galaxy's infall toward the Virgo Cluster may be a source of periodicity in cosmic-ray exposure as the solar system oscillates normally to the galactic plane, thereby inducing an observed terrestrial periodicity in biodiversity. There are a number of plausible mechanisms by which cosmic rays might affect terrestrial biodiversity. Here we investigate one of these mechanisms, the consequent ionization and dissociation in the atmosphere, resulting in changes in atmospheric chemistry, which culminate in the depletion of ozone and a resulting increase in the dangerous solar UVB flux on the ground. We estimate the enhancement of cosmic-ray intensity for a range of reasonable parameters of the galactic wind and galactic magnetic field, and use these to compute steady-state atmospheric effects. At the lower end of this range, we find that the effects are far too small to be of serious consequence. At the upper end of this range, the level of ozone depletion approaches that currently experienced due to anthropogenic effects such as accumulated chlorofluorocarbons, i.e., ∼2.1% global average loss of ozone column density. We discuss some of the possible effects. While much smaller intensity than the atmospheric effects of a nearby galactic gamma-ray burst, the duration of the effects would be about 10⁶ times greater. Current ozone depletion is a documented stress on the biosphere; it is not clear whether its consequences would be severe if of extended duration. We conclude that, for estimates at the upper end of the reasonable range of the cosmic-ray variability over geologic time, the mechanism of atmospheric ozone depletion may provide a small additional stress, enhancing the impact of other events. However, in order to account for large fluctuations in biodiversity correlated with cosmic-ray flux, other mechanisms should be investigated.