  The use of the full potential of stellar seismology is made difficult by the
improper modeling of the upper-most layers of solar-like stars and their
influence on the modeled frequencies. Our knowledge on these \emph{surface
effects} has improved thanks to the use of 3D hydrodynamical simulations but
the calculation of eigenfrequencies relies on empirical models for the
description of the Lagrangian perturbation of turbulent pressure: the
reduced-$\Gamma_1$ model (RGM) and the gas-$\Gamma_1$ model (GGM). Starting
from the fully compressible turbulence equations, we derive both the GGM and
RGM models using a closure to model the flux of turbulent kinetic energy. It is
found that both models originate from two terms: the source of turbulent
pressure due to compression produced by the oscillations and the divergence of
the flux of turbulent pressure. It is also demonstrated that they are both
compatible with the adiabatic approximation but also imply a number of
questionable assumptions mainly regarding mode physics. Among others
hypothesis, one has to neglect the Lagrangian perturbation of the dissipation
of turbulent kinetic energy into heat and the Lagrangian perturbation of
buoyancy work.
