  The squared Brunt-V\"{a}is\"{a}l\"{a} Frequency (BVF) is computed in terms of
the moist entropy potential temperature recently defined in Marquet (2011).
Both homogeneously saturated and non-saturated versions of $N^2$ (the squared
BVF) are derived. The method employed for computing these special homogeneous
cases relies on the expression of density written as a function of pressure,
total water content and specific moist entropy only. The associated
conservative variable diagrams are discussed and compared with existing ones.
Despite being obtained without any simplification, the formulations for $N^2$
remain nicely compact and are clearly linked with the squared BVF expressed in
terms of the adiabatic non-saturated and saturated lapse rates. As in previous
similar expressions, the extreme homogeneous solutions for $N^2$ are of course
different, but they are not analytically discontinuous. This allows us to
define a simple bridging expression for a single general shape of $N^2$,
depending only on the basic mean atmospheric quantities and on a transition
parameter, to be defined (or parameterized) in connection with the type of
application sought. This integrated result remains a linear combination (with
complex but purely local weights) of two terms only, namely the environmental
gradient of the moist entropy potential temperature and the environmental
gradient of the total water content. Simplified versions of the various
equations are also proposed for the case in which the moist entropy potential
temperature is approximated by a function of both so-called moist-conservative
variables of Betts (1973).
