How to define the boundaries of a convective zone and how extended is overshooting ?

Under nonlocal convection theory, convection extends without limit therefore no apparent boundary can be defined clearly as in the local theory. From the requirement of a similar structure for both local and non-local models having the same depth of con-vection zone, and taking into account the driving mechanism of turbulent convection, we argue that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is convectively unstable region when the flux is positive, and it is convective overshooting zone when the flux becomes negative. The physical picture of the overshooting zone drawn by the usual non-local mixing-length theory is not correct. In fact, convection is already sub-adiabatic (∇ < ∇ ad) far before reaching the unstable boundary; while in the overshooting zone below the convective zone, convection is sub-adiabatic and super-radiative (∇ rad < ∇ < ∇ ad). The transition between the adiabatic temperature gradient and the radiative one is continuous and smooth instead of a sudden switch. In the unstable zone the temperature gradient is approaching radiative rather than going to adiabatic. We would like to claim again that, the overshooting distance is different for different physical quantities. In a overshooting zone at deep stellar interiors, the e-folding lengths of turbulent velocity and temperature are about 0.3H P , whereas that of the velocity-temperature correlation is much shorter, being about 0.09H P. The overshooting distance in the context of stellar evolution, measured by the extent of mixing of stellar matter, should be more extended. It is estimated as large as 0.25-1.7 H p depending on the evolutionary timescale. The larger the overshooting distance, the longer the timescales. This is due to the participation of extended overshooting tail in the mixing process.