  Large-scale Rossby waves have recently been discovered from measurements of
horizontal surface and near-surface solar flows (L\"optien at al. 2018). We are
interested in understanding why only the sectoral modes are seen in the
observations and also in modelling the radial structure of the observed modes.
To do so, we characterise here the radial eigenfunctions of r modes for
slowly-rotating polytropes in uniform rotation. We find that for free-surface
boundary conditions on a spheroid of non-vanishing surface density, r modes can
only exist for $\ell=m$ spherical harmonics in the inviscid case, and we
compute their depth dependence and frequencies to leading order. For
quasi-adiabatic stratification the sectoral modes with no radial nodes are the
only modes which are almost toroidal and the depth dependence of the
corresponding horizontal motion scales as $r^m$. For all r modes except the
zero radial order sectoral ones, non-adiabatic stratification plays a crucial
role in the radial force balance. The lack of quasi-toroidal solutions when
stratification is close to neutral, except for the sectoral modes without nodes
in radius, follows from the statement that the system needs to be in both
horizontal and radial force balance. In the absence of super- or subadiabatic
stratification and viscosity, both the horizontal and radial force balances
independently determine the pressure perturbation. The only quasi-toroidal
cases in which the two determinations of the pressure perturbation are
consistent are the special cases where $\ell=m$, and the horizontal
displacement scales with $r^m$.
