## 1. Introduction

Several attempts have been published that aim to express the Brunt–Väisälä frequency (hereafter BVF) in terms of the two conservative variables represented by the total water specific content (i.e. for closed systems) and the specific moist entropy (i.e. for closed, reversible and adiabatic systems). The methods described in Durran and Klemp (1982) and Emanuel (1994)–hereafter referred to as DK82 and E94, respectively–mainly differ in the choice of ‘moist entropy’ formulation to be used as a moist conservative variable.

The entropy potential temperature *θ*_{s} recently defined in Marquet (2011, hereafter referred to as M11) corresponds to a general formulation for the specific moist entropy, valid for any parcel of moist atmosphere with varying specific content of dry air, water vapour and liquid or solid water.

The aim of this article is therefore to derive non-saturated and saturated versions of the squared BVF expressed in terms of *θ*_{s} and to compare them comprehensively with the previous moist formulations published in DK82 and E94. In this respect, the objective of the article is more general than that targeted in Geleyn and Marquet (2010), where the objective was to express the squared BVF in terms of an approximate formulation for *θ*_{s}, namely (*θ*_{s})_{1}. Comparisons between the exact and the approximate versions of the specific moist entropy defined in terms of *θ*_{s} and (*θ*_{s})_{1} will be realized with the help of the conservative variable diagrams published in Pauluis (2008, 2011; hereafter referred to as P08 and P11).

This article is organized as follows. The mathematical definition of the moist squared BVF () is presented in section 2, with expressed in terms of the gradients of the two conservative variables (*s,q*_{t}), and is established in Appendix B. The moist definition of the state equation is recalled in section 3, together with M11's specific moist entropy defined in terms of *θ*_{s}. The non-saturated and saturated versions and are then presented in sections 4 and 5, with some detailed computations available in Appendices C and D, respectively.

Several comparisons between and and the previous versions published in DK82 and E94 are presented in section 6. The comparisons are made either with the formulation expressed in terms of the lapse-rate formulation or in terms of the gradients of the two conservative variables (*s,q*_{t}), with special attention paid to the latter in section 7. The non-saturated and saturated approximate versions of the moist squared BVF expressed by (*θ*_{s})_{1} are presented in section 8.

Some numerical applications are presented in section 9, with the use of the same FIRE-I data sets as in M11. Separate analyses are made for in-cloud (saturated) and clear-air (non-saturated) air. Additional analyses are presented in Appendices E and F. First, the non-saturated moist squared BVF is compared with the usual formula expressed in terms of the vertical gradient of the virtual potential temperature. Second, the possibility of defining an analytic transition between the non-saturated and the saturated versions of the moist squared BVF is explored, using a control parameter *C* varying continuously between 0 and 1. Finally, conclusions are presented in section 10.