  We solve the nonlocal convection equations. The solutions for four model
problems are compared with results of GSPH simulations. In each case we test
two closure schemes: 1) where third moments are defined by the diffusion
approximation; and 2) where the full third moment equations are used and fourth
moments are defined by a modified form of the quasi-normal approximation. In
overshooting models, the convective flux becomes negative shortly after the
stability boundary. The negative amplitude remains small, and the temperature
gradient in the overshooting zone has nearly the radiative value. Turbulent
velocities decay by a factor of $e$ after $0.5$--$1.5\ell$, depending on the
model, where $\ell$ is the mixing length. Turbulent viscosity is more important
than negative buoyancy in decelerating overshooting fluid blobs. These
predictions are consistent with helioseismology.

Astrophysics