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Recent numerical and theoretical considerations have shown that low-degreeacoustic modes in rapidly rotating stars follow an asymptotic formula andrecent observations of pulsations in rapidly rotating delta Scuti stars seem tomatch these expectations. However, a key question is whether strong gradientsor discontinuities can adversely affect this pattern to the point of hinderingits identification. Other important questions are how rotational splittings areaffected by the 2D rotation profiles expected from baroclinic effects andwhether it is possible to probe the rotation profile using these splittings.Accordingly, we numerically calculate pulsation modes in continuous anddiscontinuous rapidly rotating models produced by the 2D ESTER (EvolutionSTEllaire en Rotation) code. This spectral multi-domain code self-consistentlycalculates the rotation profile based on baroclinic effects and allows us tointroduce discontinuities without loss of numerical accuracy. Pulsations arecalculated using an adiabatic version of the Two-dimensional OscillationProgram (TOP) code. The variational principle is used to confirm the highaccuracy of the pulsation frequencies and to derive an integral formula thatclosely matches the generalised rotational splittings, except when modes areinvolved in avoided crossings. This potentially allows us to probe the therotation profile using inverse theory. Acoustic glitch theory, applied alongthe island mode orbit deduced from ray dynamics, can correctly predict theperiodicity of the glitch frequency pattern produced by a discontinuity or theGamma1 dip related to the He II ionisation zone in some of the models. Theasymptotic frequency pattern remains sufficiently well preserved to potentiallyallow its detection in observed stars.

Oscillations of 2D ESTER models. I. The adiabatic case