  We calculate overstable convective (OsC) modes of $2M_\odot$, $4M_\odot$, and
$20M_\odot$ main sequence stars. To compute non-adiabatic OsC modes in the
core, we assume $(\nabla\cdot\vec{F}_C)^\prime=0$ as a prescription for the
approximation called frozen-in convection in pulsating stars where $\vec{F}_C$
is the convective energy flux and the prime $^\prime$ indicates Eulerian
perturbation. We find that the general properties of the OsC modes are roughly
the same as those obtained by Lee \& Saio (2020) who assumed $\delta
(\nabla\cdot\vec{F}_C)=0$, except that no OsC modes behave like inertial modes
when they tend toward complete stabilization with increasing rotation frequency
where $\delta$ indicates the Lagrangian perturbation. As the rotation frequency
of the stars increases, the OsC modes are stabilized to resonantly excite
$g$-modes in the envelope when the core rotates slightly faster than the
envelope. The frequency of the OsC modes that excite envelope $g$-modes is
approximately given by $\sigma\sim |m\Omega_c|$ in the inertial frame and hence
$\sigma_{m=-2}\approx2\sigma_{m=-1}$ where $m$ is the azimuthal wavenumber of
the modes and $\Omega_c$ is the rotation frequency of the core. We find that
the modal properties of OsC modes do not strongly depend on the mass of the
stars.
  We discuss angular momentum transport by OsC modes in resonance with envelope
$g$-modes in the main sequence stars. We suggest that angular momentum transfer
takes place from the core to the envelope and that the OsC modes may help the
stars rotate uniformly and keep the rotation frequency of the core low during
their evolution as main sequence stars.
