On the pressure equilibrium and timescales in the scale free convection theory.

Convection is one of the fundamental energy transport processes in physics and astrophysics, and its description is central to allstellar models. In the context of stellar astrophysics, the mixing length theory is the most successful approximation to handle theconvection zones inside the stars because of its simplicity and rapidity. The price to pay is the mixing length parameter that isintroduced to derive the velocity of convective elements, the temperature gradients in the convective regions and finally the energy flux carried by convection. The mixing length is a free parameter that needs to be calibrated on observational data. Pasetto et al. (2014) have proposed a new theory that determines all the properties of convective regions and the convective transport of energy with no need for a free parameter. In this study, we aim to discuss the merits of this new approach and the limits of its applicability in comparison with the mixing length theory. We present an analytical and numerical investigation of the main physical assumptions made by Pasetto et al. (2014) and compare them with the counterparts of the mixing length theory. We also present here the homogeneous isotropic limit of the Pasetto et al. (2014) theory and discuss some numerical examples to address and clarify misconceptions often associated with the new formalism.