  Within the convection zone of a rotating star, the presence of the Coriolis
force stabilizes long-wavelength convective modes. These modes, which would
have been unstable if the star lacked rotation, are called overstable
convective modes or thermal Rossby waves. We demonstrate that the Sun's
rotation rate is sufficiently rapid that the lower half of its convection zone
could possess overstable modes. Further, we present an analytic solution for
atmospheric waves that reside within a polytropic stratification. We explore in
detail the properties of the overstable and unstable wave modes that exist when
the polytrope is weakly unstable to convective overturning. Finally, we discuss
how the thermal Rossby waves that reside within the convection zone of a star
might couple with the prograde branch of the $g$ modes that are trapped within
the star's radiative zone. We suggest that such coupling might enhance the
photospheric visibility of a subset of the Sun's $g$ modes.
