  It is important to be able to calculate the moist-air entropy of the
atmosphere with precision. A potential temperature has already been defined
from the third law of thermodynamics for this purpose. However, a doubt remains
as to whether this entropy potential temperature can be represented with simple
but accurate first- or second-order approximate formulas. These approximations
are rigorously defined in this paper using mathematical arguments and numerical
adjustments to some datasets. The differentials of these approximations lead to
simple but accurate formulations for tendencies, gradients and turbulent fluxes
of the moist-air entropy. Several physical consequences based on these
approximations are described and can serve to better understand moist-air
processes (like turbulence or diabatic forcing) or properties of certain
moist-air quantities (like the static energies).
