  The only r-modes that exist in a globally barotropic, rotating, Newtonian
star are the fundamental $l = |m|$ solutions, where $l$ and $m$ are the indices
of the spherical harmonic $Y_l^m$ that describe the mode's angular dependence.
This is in stark contrast to a stellar model that is non-barotropic throughout
its interior, which hosts all the $l \geq |m|$ perturbations including radial
overtones. In reality, neutron stars are stratified with locally barotropic
regions. Therefore, we explore how stratification alters a star's ability to
support r-modes. We consider the globally stratified case and examine the
behaviour of the modes as the star gets close to barotropicity. In this limit,
we find that all but the fundamental $l = |m|$ perturbations change character
and become generic inertial modes. Restricting the analysis to $l = |m|$
perturbations, we develop the r-mode equations in order to consider stellar
models that exhibit local barotropicity. Our results for such models show that
the r-mode overtones diverge and join the inertial modes. In order to see which
r-modes persist and retain their character in realistic neutron stars, these
calculations will need to be brought into full general relativity.
