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

TL;DR

This paper proposes a new physical definition for the boundary of a convective zone based on the sign change of convective energy flux, challenging traditional views on overshooting and emphasizing the continuous nature of the transition.

Contribution

It introduces a physically motivated criterion for convective zone boundaries using flux sign change and clarifies the nature of overshooting zones in nonlocal convection theory.

Findings

Convective boundary is where the flux changes sign from positive to negative.

Overshooting zone is characterized by sub-adiabatic and super-radiative gradients.

Transition between adiabatic and radiative gradients is smooth and continuous.

Abstract

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 convection 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 ($\nabla<\nabla_{ad}$) far before reaching the unstable boundary; while in the overshooting zone below the convective zone, convection is sub-adiabatic and super-radiative ($\nabla_{rad}<\nabla<\nabla_{ad}$). 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......