## 1. Introduction

One of the conclusions of the IPCC AR4 (2007) is that cloud effects remain the largest sources of uncertainty in estimates of climate sensitivity from GCMs, with large cloud–radiative feedbacks associated with low-level clouds such as marine stratocumulus. (All acronyms, abbreviations and symbols are summarised in Appendix A.) The increase in the realism of the modelling of clouds is also one of the key features for the improvement of NWP models (both global and limited-area ones).

Different projects have already evaluated the quality of the three-dimensional distribution of clouds in climate and NWP models (e.g. EUROCS*; GCSS†). The aim of the new European Framework Program 5 EUCLIPSE‡ project is to promote the comparisons with the new space-borne remote-sensing datasets (such as CloudSat, CALIPSO, TRMM) and by realizing intercomparisons between GCM, NWP, SCM, CRM and LES outputs. The goal is to determine the main deficiencies in the parametrizations of clouds (for stratiform, shallow or deep convective ones) and to test more accurate updated schemes.

It is also possible to revisit some aspects of the theoretical concepts which form the bases of our understanding of moist atmospheric processes, such as the definition and the use of enthalpy, entropy and exergy functions. In particular, comparison with the existing *in situ* datasets could still be of some help in assessing the different hypotheses presently made to build the turbulent and convective schemes.

In this framework, the PBL region of marine stratocumuli can be considered as a paradigm of moist turbulence and it is commonly used to realize vertical diffusion of the well-known ‘conserved variables’ defined by Betts (1973, hereafter B73). However, it seems that the *in situ* observations of the Betts' variables (the liquid potential temperature and the total water content) show that these variables are not constant vertically and that the clear-air and in-cloud values are different (e.g. the vertical profiles computed with the FIRE-I dataset; De Roode and Wang, 2007, hereafter RW07).

The liquid potential temperature *θ*_{l} is defined in B73 with the aim of being a synonym for moist entropy. Therefore, *θ*_{l} may be used in moist turbulent processes as a conserved variable only if the total water content is also a constant, and these hypotheses might prevent *θ*_{l} from being a conservative quantity in case of varying dry-air and total water content, as clearly observed in the vertical profiles of stratocumulus *in situ* measurements.

One of the ways to answer these questions is to remember that, from general thermodynamics, the moist entropy must be conserved by moist, reversible and adiabatic processes (the ones acting in the moist PBL of stratocumulus). Therefore, the aim of this paper will be to compute moist entropy and its associated potential temperature as precisely as possible, and to explain how it is indeed different from, and more interesting than, the Betts' liquid potential temperature.

The use of potential temperature, instead of entropy, has a long history in meteorology and the analysis will be made in this article mainly in terms of a moist potential temperature, denoted by *θ*_{s} (with ‘s’ representing the moist entropy), in order to make the comparisons with all the existing ones easier. Nonetheless, the main variable studied in this paper is clearly the moist entropy.

The moist potential temperature, *θ*_{s}, is expected to represent all the variations of moist entropy *s*, whatever the changes in temperature, pressure, specific content of dry air, water vapour or condensed water species (solid and liquid) may be. This property would allow us to derive the same conservative properties for *θ*_{s} as the general ones valid for the moist entropy.

The concept of what is nowadays called ‘potential temperature’ in atmospheric science was first introduced by von Helmholtz (1891), with the use of the name *wärmegehalt* (warming content) and with the notation *θ*. The ‘warming content’ of a given mass of air was defined as the absolute temperature *θ* which a mass of dry air would assume if it were brought adiabatically to a normal or standard pressure. This quantity was called ‘potential temperature’ by von Bezold (1891) and the link between *θ* and the specific dry air entropy was discussed later, by Bauer (1908, 1910).

Since these pioneering studies, the concept of potential temperature has been generalized to moist air by using different approaches. The first method is to compute integrals of different approximate versions of the so-called Gibbs (1875, 1876, 1877, 1878) differential equation. With the notation of Appendix A, we can write

Equation (1) leads to the following definitions:

the liquid potential temperature

*θ*_{l}of B73, leading to a conservative moist variable, is almost constant within the stratocumulus regions if the sum of water vapour plus liquid water is a constant;the saturated equivalent potential temperature

*θ*_{ES}obtained in B73 is a companion of*θ*_{l};the ice–liquid water potential temperature

*θ*_{il}, suggested in Deardorff (1976) and derived by Tripoli and Cotton (1981, hereafter TC81), can be applied to the parametrization of the cumulus.

Another set of definitions concerns the impact of the buoyancy force, or other thermodynamic computations, leading to

the equivalent potential temperature

*θ*_{E}, obtained after the condensation level as the dry potential temperature that a parcel will have when all the water is removed from it via pseudo-adiabatic processes;the virtual potential temperature

*θ*_{v}of Lilly (1968, hereafter L68), used for instance in the thermal production term involved in the turbulent kinetic energy turbulent equations, also in the computation of the CAPE for deep convection;the liquid water virtual potential temperature

*θ*_{vl}described in Grenier and Bretherton (2001, hereafter GB01), suitable for the parametrization of the stratocumulus-top PBL entrainment.

The last method is to start with the analytic formulations for the moist specific entropy *s*, expressed as a sum of the partial specific entropies for dry air and water species. The moist potential (entropic) temperature (let us say *θ*_{s}) is then determined without the use of a Gibbs differential equation, by writing the moist entropy *s* with some prescribed reference state defined by *s*_{r}, *c*_{r} and *θ*_{sr}, leading to

Equation (2) leads to the following definitions:

different entropy temperatures in Hauf and Höller (1987, hereafter HH87), including the one denoted by in what follows (it was denoted by

*θ*_{S}in HH87);a moist potential temperature

*θ*^{∗}in Marquet (1993, hereafter M93), used in the post-processing of the ARPEGE-IFS models (subroutines PPWETPOINT and PPTHPW) and in the definition of the conservative fluxes and the barycentric equations derived in Catry*et al.*(2007);the liquid water potential temperature denoted by

*θ*_{l}in Emanuel (1994, hereafter E94), and by in what follows, including some extra terms when compared to Betts' formulation*θ*_{l}.

The paper is organized as follows. The analytic expression for the moist entropy and for *θ*_{s} will be obtained starting from the definition (2). The classical potential temperatures (*θ*_{v}, *θ*_{ES}, *θ*_{l}, *θ*_{il} and *θ*_{vl}) are first recalled in section 2. The seldom-used moist entropy potential temperatures , *θ*^{∗} and are recalled in section 3. The new formulation *θ*_{s} is then derived analytically in section 4 and in Appendix B and compared to the previous ones.

A first-order approximation for *θ*_{s} is proposed in section 5. The conservative property displayed by *θ*_{s} is computed in section 6 and compared to the one displayed by , *θ*^{∗} and .

The moist entropy potential temperature *θ*_{s} is evaluated in section 7 by using the FIRE-I experiment, with the PBL aircraft dataset described in RW07. The impacts of some of the approximations are analysed in section 8. The vertical fluxes of *θ*_{s} are computed in section 9. Paluch conserved variables and skew *T*–log *p* diagrams are analysed in sections 10 and 11 in terms of the new formulation *θ*_{s}. Some justifications of the constant feature for *θ*_{s} are suggested in section 12, including some useful Gibbs-like 3D perspectives. Finally, conclusions are presented in section 13.