  Inertial wave modes in the Sun are of interest owing to their potential to
reveal new insight into the solar interior. These predominantly
retrograde-propagating modes in the solar subsurface appear to deviate from the
thin-shell Rossby-Haurwitz model at high azimuthal orders. We present new
measurements of sectoral equatorial inertial modes at $m>15$ where the modes
appear to become progressively less retrograde compared to the canonical
Rossby-Haurwitz dispersion relation in a co-rotating frame. We use a spectral
eigenvalue solver to compute the spectrum of solar inertial modes in the
presence of differential rotation. Focussing specifically on equatorial Rossby
modes, we find that the numerically obtained mode frequencies lie along
distinct ridges, one of which lies strikingly close to the observed mode
frequencies in the Sun. We also find that the $n=0$ ridge is deflected strongly
in the retrograde direction. This suggests that the solar measurements may not
correspond to the fundamental $n=0$ Rossby-Haurwitz solutions as was initially
suspected, but to a those for a higher $n$. The numerically obtained
eigenfunctions also appear to sit deep within the convection zone -- unlike
those for the $n=0$ modes -- which differs substantially from solar
measurements and complicates inference.
