  We present an analysis of the response of a radiative region to waves
generated by a convective region of the star; this wave treatment of the
classical problem of ``overshooting'' gives extra mixing relative to the
treatment traditionally used in stellar evolutionary codes. The interface
between convectively stable and unstable regions is dynamic and nonspherical,
so that the nonturbulent material is driven into motion, even in the absence of
``penetrative overshoot.'' These motions may be described by the theory of
nonspherical stellar pulsations, and are related to motion measured by
helioseismology. Multi-dimensional numerical simulations of convective flow
show puzzling features which we explain by this simplified physical model.
Gravity waves generated at the interface are dissipated, resulting in slow
circulation and mixing seen outside the formal convection zone. The approach
may be extended to deal with rotation and composition gradients. Tests of this
description in the stellar evolution code TYCHO produce carbon stars on the
asymptotic giant branch (AGB), an isochrone age for the Hyades and three young
clusters with lithium depletion ages from brown dwarfs, and lithium and
beryllium depletion consistent with observations of the Hyades and Pleiades,
all without tuning parameters. The insight into the different contributions of
rotational and hydrodynamic mixing processes could have important implications
for realistic simulation of supernovae and other questions in stellar
evolution.
