  We explore the effects of an outer stably stratified coronal envelope on
rotating turbulent convection, differential rotation, and large-scale dynamo
action in spherical wedge models of the Sun. We solve the compressible
magnetohydrodynamic equations in a two-layer model with unstable stratification
below the surface, representing the convection zone, and a stably stratified
coronal envelope above. The interface represents a free surface. We compare our
model to models that have no coronal envelope. The presence of a coronal
envelope is found to modify the Reynolds stress and the $\Lambda$ effect
resulting in a weaker and non-cylindrical differential rotation. This is
related to the reduced latitudinal temperature variations that are caused by
and dependent on the angular velocity. Some simulations develop a near-surface
shear layer that we can relate to a sign change in the meridional Reynolds
stress term in the thermal wind balance equation. Furthermore, the presence of
a free surface changes the magnetic field evolution since the toroidal field is
concentrated closer to the surface. In all simulations, however, the migration
direction of the mean magnetic field can be explained by the Parker--Yoshimura
rule, which is consistent with earlier findings. A realistic treatment of the
upper boundary in spherical dynamo simulations is crucial for the dynamics of
the flow and magnetic field evolution.
