On a general definition of the squared Brunt-V\"{a}is\"{a}l\"{a} Frequency associated with the specific moist entropy potential temperature

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).