  Rapidly oscillating Ap stars exhibit an astrophysically interesting
combination of strong, dipolar-like magnetic fields and high-overtone p-mode
pulsations similar to the Sun. Recent time-resolved spectroscopy of these stars
unravelled a complex picture of propagating magneto-acoustic pulsation waves,
with amplitude and phase strongly changing as a function of atmospheric height.
To interpret these observations and gain a new insight into the atmospheric
dynamics of roAp stars we have carried out 2-D time-dependent, non-linear
magneto-hydrodynamical simulations of waves for a realistic atmospheric
stratification of a cool Ap star. We explore a grid of simulations in a wide
parameter space, treating oscillations of the velocity, magnetic field and
thermodynamic quantities in a self-consistent manner. Our simulations foster a
new understanding of the influence of the atmosphere and the magnetic field on
the propagation and reflection properties of magneto-acoustic waves, formation
of node surfaces, and relative variation of different quantities. Our
simulations reproduce all main features of the observed pulsational behavior of
roAp stars. We show, for the first time, that the overall depth dependence of
the pulsations in roAp atmospheres is strongly influenced by the density
inversion at the photospheric base.
