  Recent numerical and theoretical considerations have shown that low-degree
acoustic modes in rapidly rotating stars follow an asymptotic formula and
recent observations of pulsations in rapidly rotating delta Scuti stars seem to
match these expectations. However, a key question is whether strong gradients
or discontinuities can adversely affect this pattern to the point of hindering
its identification. Other important questions are how rotational splittings are
affected by the 2D rotation profiles expected from baroclinic effects and
whether it is possible to probe the rotation profile using these splittings.
Accordingly, we numerically calculate pulsation modes in continuous and
discontinuous rapidly rotating models produced by the 2D ESTER (Evolution
STEllaire en Rotation) code. This spectral multi-domain code self-consistently
calculates the rotation profile based on baroclinic effects and allows us to
introduce discontinuities without loss of numerical accuracy. Pulsations are
calculated using an adiabatic version of the Two-dimensional Oscillation
Program (TOP) code. The variational principle is used to confirm the high
accuracy of the pulsation frequencies and to derive an integral formula that
closely matches the generalised rotational splittings, except when modes are
involved in avoided crossings. This potentially allows us to probe the the
rotation profile using inverse theory. Acoustic glitch theory, applied along
the island mode orbit deduced from ray dynamics, can correctly predict the
periodicity of the glitch frequency pattern produced by a discontinuity or the
Gamma1 dip related to the He II ionisation zone in some of the models. The
asymptotic frequency pattern remains sufficiently well preserved to potentially
allow its detection in observed stars.
