  Convective overshoot mixing is a critical ingredient of stellar structure
models, but is treated in most cases by ad hoc extensions of the mixing-length
theory for convection. Advanced theories which are both more physical and
numerically treatable are needed. Convective flows in stellar interiors are
highly turbulent. This poses a number of numerical challenges for the modelling
of convection in stellar interiors. We include an effective turbulence model
into a 1D stellar evolution code in order to treat non-local effects within the
same theory. We use a turbulent convection model which relies on the solution
of second order moment equations. We implement this into a state of the art 1D
stellar evolution code. To overcome a deficit in the original form of the
model, we take the dissipation due to buoyancy waves in the overshooting zone
into account. We compute stellar models of intermediate mass main-sequence
stars between 1.5 and 8 $M_\odot$. Overshoot mixing from the convective core
and modified temperature gradients within and above it emerge naturally as a
solution of the turbulent convection model equations. For a given set of model
parameters the overshooting extent determined from the turbulent convection
model is comparable to other overshooting descriptions, the free parameters of
which had been adjusted to match observations. The relative size of the mixed
cores decreases with decreasing stellar mass without additional adjustments. We
find that the dissipation by buoyancy waves constitutes a necessary and
relevant extension of the turbulent convection model in use.
