Definition of a moist entropy potential temperature: application to FIRE-I data flights



A moist entropy potential temperature–denoted by θs–is defined analytically in terms of the specific entropy for moist air. The expression for θs is valid for a general mixing of dry air, water vapour and possible condensed water species. It displays the same conservative properties as the moist entropy, even for varying dry air or total water content. The moist formulation for θs is equal to the dry formulation θ if dry air is considered, and it displays new properties valid for the moist air cases, both saturated or unsaturated ones. Exact and approximate versions of θs are evaluated for several stratocumulus cases, in particular by using the aircraft observation datasets from the FIRE-I experiment. It appears that there is no (or only a small) jump in θs at the top of the planetary boundary layer (PBL). The mixing in moist entropy is almost complete in the PBL, with the same values observed in the clear air and the cloudy regions, including the very top of the entrainment region. The Randall–Deardorff Cloud-Top Entrainment Instability analysis may be interpreted as a mixing in moist entropy criterion. The iso-θs lines are plotted on skew T–log p and conserved variable diagrams. All these properties could suggest some hints on the use of moist entropy (or θs) in cloud modelling or in mixing processes, with the marine stratocumulus considered as a paradigm of moist turbulence. Copyright © 2011 Royal Meteorological Society

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 θl is defined in B73 with the aim of being a synonym for moist entropy. Therefore, θl 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 θl 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 θs (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, θs, 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 θs 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 image(1)

Equation (1) leads to the following definitions:

  • the liquid potential temperature θl 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 θES obtained in B73 is a companion of θl;

  • the ice–liquid water potential temperature θil, 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 θE, 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 θv 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 θvl 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 θs) is then determined without the use of a Gibbs differential equation, by writing the moist entropy s with some prescribed reference state defined by sr, cr and θsr, leading to

equation image(2)

Equation (2) leads to the following definitions:

  • different entropy temperatures in Hauf and Höller (1987, hereafter HH87), including the one denoted by equation image in what follows (it was denoted by θS 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 θl in Emanuel (1994, hereafter E94), and by equation image in what follows, including some extra terms when compared to Betts' formulation θl.

The paper is organized as follows. The analytic expression for the moist entropy and for θs will be obtained starting from the definition (2). The classical potential temperatures (θv, θES, θl, θil and θvl) are first recalled in section 2. The seldom-used moist entropy potential temperatures equation image, θ and equation image are recalled in section 3. The new formulation θs is then derived analytically in section 4 and in Appendix B and compared to the previous ones.

A first-order approximation for θs is proposed in section 5. The conservative property displayed by θs is computed in section 6 and compared to the one displayed by equation image, θ and equation image.

The moist entropy potential temperature θs 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 θs 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 θs. Some justifications of the constant feature for θs are suggested in section 12, including some useful Gibbs-like 3D perspectives. Finally, conclusions are presented in section 13.

2. The standard moist potential temperatures (B73, TC81, GB01, L68)

2.1. The versions of Betts (1973)

The potential temperatures θl and θES are defined in B73 (Eqs. (6), (7) and (9)–(12) in that paper) by approximate Gibbs differential equations and with rt assumed to be a constant. The formulation for θl and θES are obtained with several approximations, such as cpcpd and RRd, leading to

equation image(3)
equation image(4)

The corresponding values for θES and θl are obtained by integrating (4) and (3) with some further approximations (also Betts and Dugan, 1973), particularly for the last term and the variations of Lvap(T)/T with T, giving

equation image(5)
equation image(6)

Equation (6) is the equivalent of Eq. (13) in B73, expressed using the notation of Appendix A. The potential temperature (6) can be further modified by using the Taylor's series approximation exp(x) ≈ 1 + x, leading to Eq. (14) in B73 and corresponding to (7):

equation image(7)
equation image(8)

This pair of Betts moist variables (θl,qt) are nowadays used to compute the moist turbulent fluxes in most of the turbulent schemes (e.g. Brinkop and Roeckner, 1995; Cuxart et al., 2000; hereafter BR95 and CBR00)

The variables (θl,qt) are considered as conservative ones for the hydrostatic and adiabatic motion of a closed parcel of moist air, i.e. if qd = 1 − qt and qt = qv + ql are constant in the clear-air and the in-cloud regions (the precipitating species are not considered). Accordingly, the equations for the water species describe an exchange between the vapour and the liquid phases via evaporation or condensation processes, leading to

equation image(9)
equation image(10)

As already mentioned in Deardorff (1980), the conservative property is displayed for θl only if the change of Lvap(T)/T is neglected in the logarithmic derivative of (6), leading to

equation image(11)
equation image(12)

The temperature equation must be simplified too, with cp replaced by cpd (as in B73) and with 1RdT/p, leading to

equation image(13)

The expected conservative property l/dt ≈ 0 is obtained with (10) and (13) inserted into (12).

The even more simple Deardorff's (1976) formula (14) is sometimes used for θl, as in RW07:

equation image(14)

It is valid if the Exner function Π = T/ θ is approximated by 1 in the correction terms including ql (true for instance within a thin marine PBL, where θT).

2.2. The version of Tripoli and Cotton (1981).

The ice–liquid water potential temperature θil is defined by Eqs. (26) and (28) in TC81, starting from an integral of the Gibbs equation and with the same kind of approximations as in B73, with Lv and Lsub considered as constant with T and evaluated at T0. As suggested in section 4 of Deardorff (1976), θil is a three-phase generalization of θl that takes into account the impact of both rl and ri, in order to be applied to the parametrization of the liquid–ice cumulus, leading to

equation image(15)
equation image(16)

2.3. The version of Grenier and Bretherton (1981).

The liquid-water virtual potential temperature θvl is defined in GB01 (section 3-b ; Appendices A and B) in terms of the two Betts variables (7) and (8) alone.

equation image(17)

It is used in the measure of the buoyancy jump gΔi(θv)v, with the approximation Δi(θv) ≈ Δi(θvl) made in GB01 at the top of the PBL of the stratocumulus. It is also used in the computation of the PBL-top entrainment velocity (Eqs. (16), (18) and (B7) in GB01).

2.4. The version of Lilly (1968).

The virtual potential temperature θv is defined in L68 by a differential equation (Eq. (22) of that paper) and it is not based on a Gibbs equation. The aim was to seek a moist conservative thermodynamic variable, in an atmosphere subject to phase changes, which would become a measure of buoyancy. With the notation of Appendix A, and using a mean reference value equation image, this leads to

equation image(18)

The virtual potential temperature θv is not explicitly computed in Lilly (1968). It appears in the form of the vertical flux, namely equation image. Indeed, if equation image is a constant term, (18) corresponds to

equation image(19)

The vertical flux of θv defined by (19) is often used as a measure of the buoyancy fluxes, for instance in the moist thermal production equation image, one of the terms acting in the turbulent kinetic energy equations (BR95, CBR00, among others). This buoyancy potential temperature is also used for the computations of the Bougeault and Lacarrère (1989) non-local mixing length (CBR00).

It is possible to define the Lilly virtual potential temperature θv by integrating (18) with equation image considered as a constant term, then with equation image replaced by θ:

equation image(20)
equation image(21)

It can be remarked that the actual temperature associated with θv corresponds to the ‘density temperature’ denoted by Tρθv (p/p0)κ in E94.

3. The moist entropy potential temperature (HH87, M93, E94)

3.1. The version of Hauf and Höller (1987)

In HH87, the specific entropy s is defined by Eqs. (3.23) and (3.25) in terms of an entropy temperature denoted by equation image hereafter. It can be rewritten, with some algebra and with the notation of Appendix A, to give

equation image(22)
equation image(23)

As noted in HH87, this formulation for equation image supposes the existence of liquid water, at least implicitly, from the use of equation image in (22) and cl in the definition of c. This can be a drawback, since it may not be true for the most general case of an arbitrary parcel of moist air, either saturated or unsaturated, with possibly only liquid water or only ice. As for the contribution due to rl in the exponential of (23), it is a positive term contrary to what happens in θl or θil.

3.2. The Available Enthalpy version (1993)

Similarly to the method used in HH87, another moist entropy potential temperature is obtained in M93 as a by-product of the formulation for the moist exergy of an open atmospheric parcel. It is denoted by θ and, as in HH87, it is directly derived in its analytic form starting from the general formulation for s, the specific moist entropy of the system, written in M93 as

equation image(24)

It is suggested in M93 that the moist entropy potential temperatures θ and equation image be defined as

equation image(25)
equation image(26)

The interest of writing s in M93 by (24) as a complement to qd (sd)r + qt (sv)r was to avoid the problem encountered in HH87, where the definition of equation image by (22) supposes the existence of liquid water or ice, with the use of the dry air (equation image) and the liquid (equation image) standard values. On the contrary, θ defined by (24) is valid for both unsaturated conditions (rl = ri = 0) or saturated conditions (rl0 or ri0), with only the vapour reference values (sd)r and (sv)r involved, where rd and rv always exist in the atmosphere.

The exponential term in (25) is almost the same as the one of B73, at least for the common liquid water part. The difference with (6) is the term equation image, approximated by cpd in B73. It is also similar to the exponential term of TC81 recalled in (15), for both the liquid and the solid water parts. The term equation image is approximated by cpd and with Lvap(T) ≈ Lvap(T0) and Lsub(T) ≈ Lsub(T0).

Even if the purpose of HH97 was to show that modified versions of the Gibbs equation verified by equation image could lead to most of the potential temperature introduced in section 2, the entropy temperatures equation image and θ given by (23) and (25) are not directly expressed with the usual notation, as is done for θv, θl, θil or θvl. This could be one of the reasons that have prevented equation image or θ being applied in most subsequent meteorological studies.

In order to overcome this drawback, one of the purposes of the present article is to rewrite θ in a more conventional way.

The equations for the dry air and the water vapour are pd = RdρdT and e = RvρvT. The fraction pd/e is expressed in terms of

equation image

With p = pd + e, the result is

equation image(27)
equation image(28)

When (27) and (28) are inserted into (25), the terms rearrange into

equation image(29)

provided that R = Rd + rtRv, the companion of equation image.

For a dry atmosphere, rl = ri = 0 and rv tends to zero. Therefore the exponential term is equal to 1, equation image has the limit κ, equation image has the limit η and the bracketed term has the limit 1, since rv ln(ηrv) has limit 0 when rv tends to zero. As a consequence, θ has the correct dry-air limit θ.

For the moist clear-air case, rl = ri = 0, rt = rv and the exponential term is equal to 1. Nevertheless the bracketed term is different from 1 and it can impact on θ not only in cloudy regions, but also for the moist clear-air case, with θ different from the dry-air version θ.

3.3. The version of Emanuel (1994)

The liquid-water virtual potential temperature is defined in E94 starting from some approximated analytic definition of the entropy of moist air, considered as the sum of dry air, water vapour and liquid water components, with no ice content (qi = 0 and qt = qv + ql). It is assumed that

equation image(30)
equation image(31)
equation image(32)
equation image(33)
equation image(34)
equation image(35)

If (30) and (31) are exact definitions, the partial entropies equation image to equation image defined by (32)–(34) are only approximate formulae, because additional standard values should be considered, leading for instance to the correct formula equation image valid for the dry air component (with similar definitions for the water components). If T0 and p0 are set to some prescribed values, the associated standard values equation image, equation image and equation image are constant terms. However they impact on s defined by (30) or (31), not only via the possibly conservative specific contents qd and qt, but also for the non-conservative one ql. As a consequence, equation image defined by (35) is not equal to the entropy of moist air.

Nonetheless, with the use of the notation of Appendix A, when (32)–(34) are inserted into (35), equation image is defined in E94 (Eq. (4.5.15), p121) by

equation image(36)
equation image(37)

It can be remarked that 1 is denoted by ε in E94, with rt = rv + rl in R and equation image, also with equation image.

The definition (36) is different from (2), with no reference term included for the entropy or the potential temperature. It is a consequence of the approximations (32) to (34) where the reference values for the entropy are dropped.

It appears that, except the bracketed term in the second line of (37), Emanuel's formulation equation image corresponds to θ given by (29) with ri = 0. This bracketed term is an additional and arbitrary conservative quantity–i.e. only constant if rt is a true constant–introduced in E94 in order to get the formula (38), expressed with η = 1. This additional bracketed term is another reason why Emanuel's potential temperature cannot represent the moist entropy.

equation image(38)

4. The new moist entropy potential temperatures θs

The aim of the article is the same as in HH87, namely ‘to arrive at a definition of a moist potential temperature which could be regarded as a direct measure of the moist entropy’, not only for adiabatic and closed systems, but also for open systems where qd and qt are not conservative.

The problem encountered with the previous definitions for the moist entropy s, either for (22), (24) or (36), is that qd or qt appear outside the logarithm terms. They appear explicitly in (22) and (24). They are also implicitly present in (36), via c, equation image and the mixing ratio rt. The result is that equation image, equation image and θ cannot represent all the variations of the moist air entropy s if rt or qt vary.

It is possible to overcome this problem by transferring the varying specific contents qd = 1 − qt and qt inside the logarithm, and to define θs as

equation image(39)

where cpd is known and where qr, (sd)r, (sv)r and θsr are three constants to be determined.

The computation of the quotient θssr is presented in Appendix B. It is suggested that θs be defined as

equation image(40)

where the reference potential temperature is written

equation image(41)

The term (1 + ηrr)κ is different from 1 and it is put into (41)–instead of (40)–in order to fulfil the demand that θs must be equal to θ for the dry-air case (i.e. for qt = qv = ql = qi = 0 and qd = 1). The term exp(Λqr) appears in (41) in order to verify the expected property θsr = θs(Tr,pr,qr; ql = qi = 0), which results from the choice of (1 − qr)(sd)r + qr (sv)r as a reference entropy in (39), balancing the term exp(Λqr) in (41).

Contrary to the potential temperatures θil, equation image, θ and equation image where only the mixing ratios are involved, the formula (40) for θs is written in terms of the specific contents in the exponential term, as for θv, θl and θvl.

The main difference from all other formulations is the term exp(Λqt) in (40), with Λ = [(sv)r − (sd)r]/cpd. Thus it is necessary to deal with the difference in the absolute values for the dry air and the water vapour reference partial entropies defined in HH87.

The value for s in (39) is independent of any arbitrary choice for the reference temperature Tr, pressures pr = (pd)r + er and specific contents (qd)r = 1 − qr. However, another choice for Tr, er and qr would modify the reference values (sv)r, (sd)r and θsr, and also θs in (40). As a consequence, it will be important to choose the reference values accurately, so that the variations of θs with T, p, qv, ql and qi could be similar to the equivalent variations of s. (See the sensitivity experiments presented at the end of section 8.)

The reference entropies (sv)r and (sd)r are determined at the temperature Tr and at the partial pressures (pd)r = prer and er. They are computed from the standard values equation image and equation image (Appendix A), by the use of the equivalent of (B.8) and (B.9), yielding

equation image(42)
equation image(43)

5. First-order approximations for θs

For practical purposes, it would be interesting to write a simple version for (40), with the use of the first-order approximation exp(x) ≈ 1 + x valid for the exponential terms and for small values of x, as used before to derive (7) from (6) in B73 and (16) from (15) in TC81. The other power terms of the form ab will be rewritten as exp[bln(a)] and they will be approximated by 1 + bln(a) for small values of bln(a). All the products (1 + x)(1 + y) will be approximated by 1 + x + y for small values of x and y (for instance qv, qt or qr), with second-order terms like x y discarded.

As indicated in (44), the moist entropy potential temperature θs can be written as a sum of two terms:

equation image(44)

The first term (θs)1 is given by the first line of (40), leading to the expressions (45) to (47). It will be shown in section 8 that (θs)1 is indeed the leading-order term of θs for the stratocumulus cases in FIRE-I.

equation image(45)
equation image(46)
equation image(47)

The second term (θs)2 is given by (48). It is derived from a leading-order approximation of the remaining part of (40), i.e. the second and third lines, valid for small values of qt and rv.

equation image(48)

First-order approximate versions of (45) to (47) are obtained with exp(x) ≈ 1 + x, leading to

equation image(49)
equation image(50)

All the formulae (45)–(47), (49) or (50) valid for the first term (θs)1 contain the term Λqt. It is an extra term compared with the liquid-water (B73) and the ice-liquid water potential temperatures (TC01), recalled in (6), (7), (15) and (16). The ice component Lsub(T)qi is the logical complement to B73's formula, with the latent heat Lvap and Lsub expressed for the actual temperature T, and not at T0 as in the TC01's formula. Also, the formulations (45), (47) or (50) are similar to GB01's formulation (17), with δ replaced by Λ.

The formula (48) for the second term (θs)2 can always be computed because both −γ ln(rv/rr)qt and −κδ ln(p/pr)qt has limit zero as qt tends to zero, providing that qt decreases more rapidly than ln(rv) and ln(p).

6. The conservative properties displayed by θs

The three entropy potential temperatures equation image, θ, and θs display conservative properties if qd, qt and rt are constant, whatever the possible reversible exchanges existing between the vapour, liquid or solid water species may be.

These properties are not easy to prove starting directly from (23), (29) or (40), where changes in rl and ri must be carefully analysed. It is much easier to analyse the corresponding moist entropy definitions (22), (24) or (39), because all the terms except equation image, θ and θs depend only on qt or rt, which results in a partial conservative feature for the moist potential temperatures, only valid for constant values of qt and rt and only if the moist entropy s is a constant for adiabatic and reversible processes occurring within a closed parcel of fluid.

The same partial conservative property is displayed by equation image if qd, qt and rt are constant. Even if (37) is based in E94 on the approximate moist entropy equation image given by (35), different from the true moist entropy (30), the definition (37) for equation image only differs from the one (29) for θ by the aforementioned bracketed term in the second line of (37), and this bracketed term depends only on rt, from which the same partial conservative property holds for equation image.

A more general conservative property is displayed by θs for a region where the entropy is well-mixed, either by diffusion, turbulent, convective or dynamical processes. In that case, for constant values of s given by (39), θs defined by (40) is also a constant even if qd, qt and rt vary in the vertical or in the horizontal. A more precise analysis is derived in the Appendix C.

7. Numerical evaluations: the FIRE-I dataset

7.1. The entropy for flights RF03B, 02B, 04B, 08B

The exact and approximate versions of θs are analysed with the aircraft observations of the stratocumulus boundary layer during the First ISCCP Regional Experiment (FIRE-I), performed off the coast of southern California in July 1987. As in RW07, ‘the mean values computed from the aircraft data may be loosely interpreted as typical grid-box mean values in a general circulation model and the standard deviation as a measure of the sub-grid variability’.

The aircraft measurements of the temperature, the water vapour concentrations and the liquid water content are not local ones. They are sampled and averaged over at least ≈ 100 m during the radial flights However, these sampling aircraft observations will be considered as ‘local’ measures hereafter, for the temperature and the specific contents (water vapour and liquid water). The local measures are conditionally averaged in this study following RW07's method, by separating the in-cloud from the clear-air conditions with the threshold ql > 0.01 g kg−1.

As in RW07, the average values are computed within fixed height intervals with a depth Δz = 25 m. Unknown instrumental errors impact on the accuracy of all the data. It has been decided to correct two of them, with partial removal of the supersaturated or unsaturated in-cloud regions. The water vapour specific content qv will be modified if the measured liquid water is above a critical value (ql)c. In that case qv is set to its saturation value qws(T) (J. L. Brenguier, personal communication). It is also ensured that qvqws(T). These corrections may have not been done in RW07 and they can explain the small differences from the RW07 results. Another difference with RW07 is the use of the exact definition for the Betts potential temperature (6) in the present study, whereas Deardorff's formulation (14) is used in RW07.

According to several tests discussed at the end of section 8, the reference values have been set to Tr = T0 = 273.15 K, pr = p0 = 1000 hPa, er = esw(T0) ≈ 6.11 hPa and rrεer/(prer) ≈ 3.82 g kg−1. The corresponding constant Λ ≈ 5.87 is obtained with (sv)r and (sd)r given by (42) and (43).

The Δz = 25 m average values of the moist entropy equation image are depicted for the flight RF03B in Figure (1). They are evaluated from (39), with θs and θsr given by (40) and (41) and with the averaging operator derived in the Appendix D.

Figure 1.

The in-cloud (dark square) and the clear-air (open square) vertical profiles for the average moist entropy equation image, depicted for the flight RF03B (2 July 1987). The large box (heavy line) represent the cloud region and the smaller box (thin line) the PBL-top entrainment zone, with the same definitions and values for the PBL-top height and the free-air base height as published in RW07. The horizontal bars indicate one standard deviation from the mean values, with small vertical lines at the end of the in-cloud bars. Other information (on Δθ = 1 K, and on the sketch profile denoted by thin solid segment lines) is available in the text.

The important result is that, for a given level, the clear-air and the in-cloud values have the same moist entropy, with the standard deviations of the two conditionally averaged subsets crossing over. Moreover, the moist entropy is almost constant up to 1050 m or so, including the entrainment region.

In order to make the comparison easier with the usual jumps in θl of more than 8 to 10 K, a width Δθ = 1 K is plotted, indicating the small impact on entropy associated with a change in potential temperature from 300 to 301 K, leading to cpd ln(301/300) ≈ 3.34 J K−1kg−1.

It appears that the entrainment region is characterized by the largest standard deviations of the PBL, for both the clear-air and the in-cloud conditions. It could be interpreted as an increase in the sub-grid variability for equation image with a possible partial mixing in moist entropy in the entrainment region, where the moist PBL air and the dry air above entrain or possibly detrain (RW07).

A series of (solid) line segments are plotted in Figure (1). They form a sketch profile for equation image, with a constant value of 6884 J K−1kg−1 plotted up to 950 m corresponding to a full mixing of equation image within the PBL. It is observed that the PBL-top mixing is realized with no obvious inversion jump in moist entropy, or possibly a small jump of less than 1.5 K in potential temperature. There is a linear trend above the PBL-top height (1025 m), due to the impact of the radiation and subsidence processes.

All these results suggest that the moist PBL is homogeneous in equation image, with a continuous transition with the dry air above. As a consequence, equation image could be an interesting candidate for being a true conservative variable to be used somehow in atmospheric turbulent schemes, where no vertical mixing in equation image may result in zero turbulent tendencies (for all the clear-air, in-cloud or grid-cell average parts).

The properties suggested by the analyses of the moist entropy computed for flight RF03B can be strengthened by the same analyses applied to the three other flights, as shown in Figure 2. Even if the differences in the average values for the clear-air and the in-cloud subsets are larger in the PBL-top entrainment regions for the flights RF04B and RF08B, the average values of one subset are located within the horizontal bars of the other. The conclusion is that the clear-air and the in-cloud subsets seems to have almost the same moist entropy for all the FIRE-I data flights, with a common value for equation image almost constant within the PBL and with a smooth transition occurring with the dry subsiding air located above the PBL regions.

Figure 2.

As Figure 1, but for (a) flight RF02B (30 June 1987), (b) flight RF04B (5 July 1987) and (c) flight RF08B (14 July 1987).

7.2. Other parameters for flight RF03B

The average values for the two moist potential temperatures l> and <(θs)1>, the specific total water contents equation image and the liquid water content equation image are depicted in Figure 3 for the flight RF03B. The values of θl and (θs)1 are computed with the exponential expressions (6) and (47), respectively. Figure (3)(c) for the liquid water content shows that RF03B documents a thin layer of homogeneous stratocumulus.

Figure 3.

For flight RF03B (2 July 1987), with in-cloud data shown as dark circles or squares and clear-air data as open circles or squares. (a) The mean values for the moist potential temperatures l> (left) and <(θs)1> (right), with θl and (θs)1 computed with (6) and (47). (b) The mean values for the total water specific contents equation image. (c) The mean values for the liquid water specific content equation image. In (c), the threshold (ql)c is represented by a vertical dashed line, above which qv is set to its saturation value qsw(T). Boxes and horizontal bars are as in Figure 1.

The shape of the vertical profiles of <(θs)1> in Figure 3(a) is close to the one observed for equation image in Figure 1. It confirms that, at least for this case and for the aforementioned set of reference values, <(θs)1> is indeed a relevant synonym for equation image. It is not true for B73's mean values l> and equation image in Figures 3(a) and (b) respectively, for which linear trends exist in the PBL (+1 K and −1 g kg−1 from the surface to 850 m, and even much larger changes in the cloud and the entrainment region).

Large values are observed for the differences in l> between the clear-air and the in-cloud regions, denoted by Δl>. They increase with height, reaching about 4 K in the entrainment region, as indicated in Figure 3(a). There is an associated decrease with height of equation image in the entrainment region, with equation image g kg−1 at the top of the entrainment region, as indicated in Figure 3(b).

The clear-air values of l> are 4 K higher than the in-cloud ones. They lead to a difference of 1.3% or so. The term exp(Λ qt) corresponds to an opposite impact of the order of −1.2%. Since the liquid water term equation image depicted in Figure 3(c) gives the same contribution for l> as for <(θs)1>, the almost opposite numerical impacts of ±1.2% explain how the new term exp(Λ qt) acts in (45)–(47) in order to make <(θs)1 > constant with height and to give the same clear-air and in-cloud values.

Large jumps in l> and equation image are observed within the entrainment region in Figure 3(a) and (b). They are in agreement with the values indicated in RW07 for this flight (10.1 K and −4.9 g kg−1). As for equation image or <(θs)1>, the entrainment region is characterized for l> and equation image by larger standard deviations and may be interpreted as an increase in sub-grid variability.

The jump in <(θs)1> is much smaller than the one for l> (i.e. 1–2 K versus 10.1 K), or possibly does not exist.

For a given level, the standard deviation bars of the clear-air and in-cloud conditionally averaged subsets do not cross over for l> and equation image. It seems that the clear-air and the in-cloud values cannot be considered as equal for l> and equation image, in contrast with the result obtained with equation image and <(θs)1>.

7.3. All parameters for flights RF02B, 04B, 08B

Other computations made for flights RF02B, RF04B and RF08B are presented in Figures 4–6. The clear-air and the in-cloud values of equation image, equation image and equation image are similar to the corresponding results shown in RW07. The panels (c) for the liquid water content show that RF04B represents a thin layer of heterogeneous stratocumulus, whereas RF02B and RF08B represent thick layers and rather heterogeneous clouds (liquid water exists in almost the whole PBL).

Figure 4.

As Figure 3, but for flight RF02B (30 June 1987).

Figure 5.

As Figure 3, but for flight RF04B (5 July 1987).

Figure 6.

As Figure 3, but for flight RF08B (14 July 1987).

The same properties observed for the flight RF03B are confirmes by the others. In particular, the vertical profiles of <(θs)1> are almost constant within the whole PBL, including the entrainment regions, especially for the flight RF08B. Also, in contrast with the large differences observed with equation image , the values for <(θs)1> are almost equal in clear-air and in-cloud conditions, with the same impact found for the term exp(Λqt) for the three flights. The impacts are ±1.7% for RF02B, ±2.3% for RF08B, and a partial balance of +2.7% and −2.1% for flight RF04B (however the standard deviations of the two conditionally averaged subsets also cross over for this latter flight, indicating that the difference may not be significant).

It can be noted that the standard deviations in the clear air above the PBL top are much larger for <(θs)1> than for l> for the flight RF04B. It is an impact of the high level of sub-grid variability existing for equation image in this flight, with an influence on <(θs)1> only and with no impact on l>.

The variation with height of <(θs)1> for flights RF04B and RF02B and above the PBL top is more complex than for RF03B. The vertical gradients of <(θs)1> are largely influenced (or may be dominated) by the vertical gradients of equation image. The almost constant values for equation image depicted for the flights RF03B and RF08B above the PBL top can explain the linear positive trend observed for these flights, where the increase in <(θs)1> follows the increase in <θ>.

7.4. The grid-cell mean values

The grid-cell mean values for l> and <(θs)1> are depicted in Figures 7 (a)–(d), for the four radial flights. The grid-cell values represent the internal variables available in NWP models, either GCMs or SCMs.

Figure 7.

The grid-cell mean values of the moist potential temperatures l> (on the left, open circles) and <(θs)1 > (on the right, open squares) for flights (a) RF02B, (b) FR03B, (c) RF04B, and (d) RF08B. Boxes are as in Figure 1.

The computations of the grid-cell average values are more relevant for the moist entropy–or for <(θs)1>–than for l>, because the in-cloud and the clear-air values are equal only for <(θs)1>, not for l>.

The other properties observed for the in-cloud and clear-air averages are also valid for the grid-cell averages. The jumps in l> within the entrainment region are large and they correspond to the expected results already published for these FIRE-I cases (e.g. RW07). On the contrary, the jump in <(θs)1 > does not exist, and it is possible to assess the constant value for the grid-cell average of <(θs)1 > up to the PBL top, with the constant value also valid in the entrainment region since it is located within the horizontal bars, no more than one standard deviation from the mean values.

There is higher sub-grid variability for <(θs)1> in the entrainment region for all flights. The sub-grid variability is also larger in the dryer air above the PBL top for flight RF04B, due to an especially high sub-grid variability for qv for that flight (Figure 5(b)).

In order to be more confident in the previous results (i.e. constant PBL values and no jump in <(θs)1>), it is interesting to somehow quantify the impact of the instrumental or measurement errors on <(θs)1>. It is possible to use a Monte-Carlo method by adding a series of perturbations to the original data flight values. For each of the basic variables (θ, qv, ql), the sets of perturbations are defined by (±0.1%, ±2%, ±5%) for the weak ones and (±0.3%, ±5%, ±10%) for the strong ones. The constraint qv< qsw is still fulfilled and it can prevent some of the perturbations in qv. The weighting factors are arbitrarily set to 75% for the original data, 20% for the small perturbations and 5% for the higher ones.

The result is depicted in Figure 8 where the horizontal bars represent the global impact of both the Monte-Carlo perturbations and the sub-grid variability. The mean vertical profile of <(θs)1> is the same as in Figure 1. The only difference is that the horizontal bars are larger due to the Monte-Carlo perturbations. The hypothesis of a constant value for the moist entropy (6884 J K−1kg−1) is better supported than in Figure 1, for all levels located within the PBL up to 1025 m and for both the clear-air and the in-cloud regions.

Figure 8.

As Figure 1, but with the Monte-Carlo perturbations added for θ, qv and ql.

7.5. The links between Δ<(θs)1 >= 0, CTEI and the (Δl>, equation image) plane

The differences between the clear-air and the in-cloud values for l> and equation image are denoted by positive values for Δl> and negative values for equation image. They have been computed for the four FIRE-I flights (02B, 03B, 04B, 08B) and for the few highest in-cloud levels located within the entrainment regions (from 4 to 11 points, depending on the flights). The resulting (Δl>, equation image) plane is depicted in Figure 9.

Figure 9.

A plot of the differences between clear-air and in-cloud values for the mean Betts variables l> and equation image. The points of coordinates (X = Δl>, equation image) are plotted for all the Δz = 25 m average layers located within the entrainment regions (thin line boxes in Figures 36) for flights RF02B (open circle), RF03B (dark square), RF04B (open diamond) and RF08B (dark triangle). The solid line represents the least-squares fitted curve. The dashed line represents the ‘moist isentropic’ curve for which Δ[<(θs)1 >] = 0, with positive values above the dashed line and negative values below (Λ = 5.87; θ ≈ 300 K).

The reason why the usual jumps in θl and qt across the cloud-top capping inversion are not used is that these jumps are defined with poor accuracy, depending on the definition of the free-air base level (RW07). On the contrary, the differences between the clear-air and the in-cloud values are unambiguous. They are defined for each level and the clear-air values are typical of the air located above the inversion, whereas the in-cloud values are typical of the moist PBL values, leading to a difference computed locally at each level which is typical of the ‘jump accros the cloud-top capping inversion’.

With <(θs)1> approximated by (49), the differences between clear-air (cl) and in-cloud (in) values are

equation image(51)
equation image(52)

The term in the second line of (51) is neglected in (52), with (θ)in replaced by θ.

For θE approximated by (53), it is possible to express the differences in <(θs)1> as (55), if the differences in equivalent potential temperature are given by (54).

equation image(53)
equation image(54)
equation image(55)

The slope of the fitted line in Figure 9 is equal to −2406 K (kg/kg)−1. It corresponds to a value for Λ that would make the clear-air and the in-cloud values equal in terms of moist entropy, leading to Δ<(θs)1>= 0 in (52) and to a slope equal to Λθ. For θ ≈ 300 K, this gives Λ = 2406 ≈ 8. This value is higher than Λ = 5.87 obtained with (sv)r and (sd)r given by (42) and (43). The explanation for this difference is that Δ<(θs)1> is not exactly equal to zero in (52) and in the entrainment regions of the four FIRE-I flights (even if the mean values are located within the error bars of the others).

The dashed line depicted in Figure 9 corresponds to a MIME, where the clear-air and the in-cloud values of <(θs)1> are equal. It seems that this dashed line looks like the ‘cloud-top instability criterion’ proposed by Randall (1980) and Deardorff (1980), also called ‘buoyancy reversal criterion’ or CTEI. The CTEI line is depicted as Δ2 = 0 in WR07, with a plot of the points corresponding to the jump across the inversion for the four FIRE-I flights (02B, 03B, 04B, 08B).

It is possible to interpret the CTEI line differently, in terms of a MIME (i.e. with the same values for the potential temperature (θs)1 above the cloud and for the in-cloud and the clear-air subparts of the entrainment region). From (55) and (52), the hypothesis Δ<(θs)1>= 0 corresponds to the straight lines defined by

equation image(56)
equation image(57)

According to Yamagushi and Randall (2008), the ‘cloud-top instability criterion’ proposed by Randall (1980) and Deardorff (1980) corresponds to (56). As suggested by Lilly (2002), the CTEI analysis can also be realized with the help of (57). Depending on the chosen plane, the CTEI slopes are written either as

equation image

or as

equation image

The link between the two parameters kRD and kL and the MIME slope Λ θ given by (57) is

equation image(58)

The CTEI criterion parameter kRD has the standard value of 0.23 in Kuo and Schubert (1988). It is mentioned in Yamagushi and Randall (2008) that kRD must vary from 0.18 to 0.48 with the mean potential temperature of the PBL θ varying from 275 to 325 K. MacVean and Mason (1990) have derived different values, depending on the saturated or unsaturated conditions observed for the above-cloud and in-cloud conditions: 0.23 for saturated/saturated (the Randall–Deardorff value) and 0.70 for unsaturated/saturated (the more relevant situation). Lilly (2002) has derived a real situation value of kRD = 0.61 (for kL = 2.55), with the standard value kRD = 0.22 obtained as a limit case for kL = 1.28. From the (Δl>, equation image) plane published in RW07 and Duynkerke et al. (2004), kRD are set to 0.26 and 0.18, respectively.

From the relation (58), the value Λ ≈ 5.87 retained in this paper and the mean condition θ ≈ 300 K valid for the FIRE-I datasets lead to kRD = 0.29. This value corresponds to a MIME criterion and it compares with the previous values obtained in the studies of the CTEI criterion (varying from 0.18 to 0.70).

8. Sensitivity experiments

The first test depicted in Figure 10(a) and (b) concerns the evaluation of the error between the approximate version <(θs)1 > and the exact one s>. There is a small negative bias of −0.35 to −0.55 K, which corresponds to an error of less than 0.2%. It justifies the use of <(θs)1 > in the previous analyses.

Figure 10.

Sensitivity experiments. (a) The RF04B profiles of exact values s> (heavy line) and the corresponding leading-order approximate formulation <(θs)1> (thin line). (b) The RF04B profile of the difference between <(θs)1> and s>. (c) The impact on (θs)1 of the threshold value (ql)c above which qv is set to its saturation value qsw(T), with the RF04B regular values shifted by the amount −8 K on the left and the RF04B modified values located on the right. The two vertical dashed lines are shifted by the same amount of −8 K, in order to simplify the comparisons. (d) The whole RF03B dataset extended above the PBL to 2800 m, for both l> (thin black line) and <(θs)1 > (heavy black line). Thin white line segments are plotted over the vertical profile of <(θs)1>, indicating a possible linearised description of it. Boxes and the standard deviation bars are as in Figure 1.

The second test is shown in Figure 10(c). It corresponds to the impact of the threshold value (ql)c on the clear-air and in-cloud values of (θs)1, as described in section 7.1. According to Figure 5(c), (ql)c = 0.04 g kg−1 for flight RF04B and the possible impacts could only concern the upper in-cloud levels located between 850 and 975 m height, for which equation image. It appears that the modified in-cloud values get closer to the clear-air ones for the layers 925–950 m and 950--975 m, with the horizontal bars crossing over. It justifies the use of qv = qsw(T) where qv > (ql)c locally.

The third test concerns the analysis of the full vertical range for the flight RF03B, including the extended levels reaching 2800 m and above. The aim is to check whether or not the vertical profile of the approximated new potential temperature <(θs)1> exhibits a standard stable layer pattern far above the PBL. It appears that the ‘stable linear regime’ already depicted as solid line segments in Figure 1 can be extended above the PBL, as suggested for the grid-cell average depicted in Figure 10(d) as white solid segments.

As a consequence, it may be more relevant to search for a description by line segment starting with the vertical profiles of equation image or <(θs)1>, rather than with the vertical profiles of l>. Applications could be found in the building of idealized initial profiles as used in the SCM, CRM or LES intercomparison cases.

Another set of tests is shown in Figure 11(a) and (b), where grid-cell average values have been computed for flight RF03B and for all the potential temperatures described in sections 2 and 3. It appears that the TC81 and E94 values for θil and equation image are very close to the Betts θl. The (buoyancy) virtual potential temperatures θv (L68) and θvl (GB81) are 2 K higher than the Betts-like ones. The same is true for the entropy potential temperature equation image (HH87).

Figure 11.

Vertical profiles for the grid-cell averages of several potential temperatures, for the flight RF03B. (a) Comparison of (θs)1 with (from left to right) θil (TC81), θvl (GB01) and θv (L68). (b) Comparison of (θs)1 with (at left) equation image (E94), equation image (HH87) and θ (M93). The four profiles located on the right of (b) show (from left to right) the four θE formulations of B73, E94, Bolton (1980)–Eqs. (21) and (43)–and a numerical computation made with a code developed by J.M. Piriou from the ARPEGE model. (c) The impact on (θs)1 of different choices for Tr of 250, 273.15, 278 and 320 K. Boxes are as in Figure 1.

The profile for <(θs)1> in Figure 11(b) is different from all others, with a difference of more than 14 K from the Betts-like or virtual potential temperatures and with the moist available enthalpy potential temperature θ leading to values between. Clearly, θl cannot represent the moist entropy.

The warmest profiles on the right of Figure 11(b) allow a comparison between <(θs)1> and four different formulations for the equivalent potential temperature. The coldest profile for E> is based on the simplified formulation (53), with (49) representing the first-order expression for (45). The comparison of (49) with (53) explains the reason why the vertical profile of <(θs)1> is roughly two-thirds of the way between equation image and θE, with Lvap/(cpdT) and Λ indeed close to 9 and 6, respectively.

As a consequence, it seems that the moist entropy equation image and the associated moist potential temperatures s> and <(θs)1> cannot be represented by any of the other potential temperatures.

The last test concerns the choice of the reference potential temperature Tr. The variations of Λ with Tr and pr are presented in Table I.

Table I. The values for Λ = {(sv)r − (sd)r}/cpd given as a function of Tr (K) and pr (hPa). The values pr = 1000/exp(1) ≈ 368 hPa and Tr = 250 K were used in M93. The bold value Λ = 5.87 corresponds to pr = 1000 hPa and Tr = 273.15 K, as retained in the present study.

The sensitivity associated with changes in Tr is more important than with changes in pr. The value Λ = 5.87 corresponds to the special choices for Tr and pr indicated in Appendix A.

Four profiles are depicted for flight RF03B in Figure 11(c), corresponding to the grid-cell average of θs and for Tr = 250, 273.15, 278 and 320 K. One of the rules for choosing a relevant correct value for Tr is to search for the ‘same vertical profile’ for <(θs)1> in Figure 11(c) as in Figure 1 for the vertical profile of the moist entropy equation image. It is also useful to compare the vertical profiles for <(θs)1 > and for equation image for the three other flights, as described in Figure 2 for the entropy and Figures 46 for the corresponding potential temperatures.

It seems that the values Tr = 273.15 K (chosen in the present study) or Tr = 278 K are appropriate ones, at least for these FIRE-I flights. It can be noted that the change in <(θs)1> is less than ±3 K in the PBL, even for the extreme variations of Tr from 250 to 320 K, and it is less than ±1 K above the PBL. These changes may be considered as small in comparison with the large differences between <(θs)1> and the other potential temperatures, as depicted in Figures 11(a) and (b).

In spite of these encouraging sensitivity experiments, one may consider that for global applications of (θs)1 in GCMs (or in NWP models with sufficiently large horizontal domains), whether it may be difficult to find a value for Tr (and for Λ) which may be relevant for all points, going from equatorial to polar regions.

However it is important to remember that (θs)1 is only the first-order approximation of the exact formulation (40), and it can be verified that the numerical values for the exact moist entropy s and the moist potential temperature θs do not depend at all on Tr or pr, as indicated in Table II. The large changes in the two terms sr and θsr balance each other in order to give constant values for the exact potential temperature θs and for the reference entropy sref, with sref defined by

equation image(59)
equation image(60)
equation image(61)

The quantity sref can be evaluated with (42), (61) and (B.16) inserted into (60), leading to

equation image

for the standard values of equation image and T0 given in Appendix A.

Table II. Numerical values computed for the same parcel of cloud (p = 800 hPa, T = 280 K, qv = 7.74 g kg−1, ql = 1 g kg−1, qi = 0) but with different values of Tr (K) and pr (hPa). From (39), the moist entropy is equal to s = sr + cpd ln(θssr), with sr given by (61). The moist potential temperatures θs, θsr and (θs)1 are given by (40), (41) and (45). The reference entropy sref is defined by (60).

The formula (59), where cpd and sref are equal to two thermodynamic constants, demonstrates that θs is a true synonym of the moist entropy. The consequence is that the analysis of the vertical profiles of s or θs can be realized with no approximation, whatever the choices for Tr and pr may be.

If an approximate version of θs is needed, the bold values of (θs)1 presented in Table II show that 273.15 K is a relevant value for Tr, with a negative bias in the computation of (θs)1 less than 1 K, and corresponding to the values depicted in Figure 10(a) and (b).

9. Vertical fluxes of θs

According to the formulation (39), the moist entropy depends on the logarithm of θs. It is approximated by the logarithm of (θs)1 given by (45), leading to

equation image(62)

The differential of (62) is written

equation image(63)

and the flux of moist entropy is then approximated by

equation image(64)
equation image(65)

The flux of (θs)1 is written

equation image(66)

If the moist entropy is a constant within the PBL–as observed for the FIRE-I flights–then equation image and, from (64), equation image. When this assumption is introduced into (66), it leads to a moist isentropic balance of the Betts variable fluxes and, according to (45), it is written as

equation image(67)

This relation between the Betts variable fluxes correspond to the CTEI criterion and to (57).

In some parametrizations of the turbulence, the internal variables used in the numerical schemes are based on a modified static stability function defined by cpdT + gzLvapql. It replaces the use of θl. The trick is to take into account the hydrostatic (exact) differential and (approximate) flux equations

equation image(68)
equation image(69)

and to use the original Betts formula (6), with the variations of Lvap(T)/T with T neglected with respect to the changes in ql, to arrive at

equation image(70)

where the liquid water static energy Sl is defined in Stevens et al. (2003) by

equation image(71)

The flux of moist entropy is then obtained with (70) inserted into (65) and (64), leading to

equation image(72)

where Sm is the perturbation of a kind of ‘moist entropy static energy’ function Sm defined by

equation image(73)

or equivalently by

equation image(74)

In comparison with the liquid water static energy (71), Sm given by (73) contains the additional component equation image. This term is not constant with height if equation image varies with z, even if qt is a constant (as an invariant of the moist system). Only the moist entropy flux (72) is a constant, including the division by equation image. It is the reason why the quantity equation image is plotted by Stevens et al. (2003) in place of θl, corresponding to the flux (70).

The additional part between equation image given by (73) and equation image given by (71) is cpd Λqt. It can only be discarded if qt is a constant, a property not verified in the entrainment region, where possible large differences could exist between the flux of equation image and the flux of equation image.

10. Other stratocumulus cases; conserved variable diagram

To obtain a more general appreciation of the value of the moist entropy–or (θs)1–in atmospheric science, three well-known stratocumulus cases have been analyzed from different published papers, representing different regions and times.

The northeastern Atlantic Ocean ASTEX profiles (June 1992) are plotted for (θl, qt) in Cuijpers and Bechtold (1995), the southeastern Pacific Ocean EPIC profiles (6-day mean values, October 2001) are plotted for (θ, qv, ρql) in Bretherton et al. (2004), and the northeastern Pacific Ocean DYCOMS-II profiles (RF01 dataset, July 2001) are plotted for (θ, qt, ql) in Zhu et al. (2005).

The vertical profiles of θl and (θs)1 are plotted for the three cases in Figure 12. As for the grid-cell values of the FIRE-I cases depicted in Figure 7, there is no (EPIC, DYCOMS-II) or only a small (ASTEX) jump in moist entropy potential temperature at the top of the PBL, with (θs)1 a constant throughout the PBL in the three cases.

Figure 12.

The Betts and moist entropy potential temperatures for (a) the ASTEX, (b) the EPIC and (c) the DYCOMS-II (RF01) stratocumulus cases. The liquid-water potential temperature θl is depicted by solid lines, (θs)1 by dashed lines.

In the conserved variable diagrams, the total specific content of water vapour is plotted against the equivalent potential temperature (Paluch, 1979) or the liquid-water potential temperature (Neggers et al., 2002). Figure (13) is the (qtl) diagram for the four FIRE-I data flights and for the three other stratocumulus cases ASTEX, EPIC and DYCOMS-II (RF01). This diagram can be used as a graphical method to demonstrate (or to appreciate) the constant moist entropy regime and the MIME processes occurring within the PBL of these stratocumulus cases.

Figure 13.

A conserved variable diagram with the total specific content equation image plotted against the liquid water potential temperature l>. The four FIRE-I data flights (02B, 03B, 04B and 08B) are represented together with the three EPIC, ASTEX and DYCOMS-II (RF01) datasets. The entrainment regions are depicted between the heavy dashed lines, with the free upper-air points located above 300 K on the bottom right and the moist PBL points grouped on the left. The slanting grey solid lines correspond to constant values of (θs)1 (they can be labelled at 6 K intervals with values of θl at qt = 0).

The moist PBL values are assembled on the left of the diagram, with small increases in θl with height and associated decreases in qt. The upward variation of the points in the PBL and then in the entrainment regions correspond to changes along slantwise patterns approximately following the constant (θs)1 lines, defined by Λqt = ln{(θs)1l}. Clearly, from the left to the right there are constant regime or smooth transitions for all flights in terms of (θs)1 between the moist PBL, the entrainment region and the free upper air, where (θs)1 starts to increase due to the diabatic heating processes and to the subsidence of the dry air located above.

The ASTEX curve depicted in the conserved variable diagram (Figure 13) is different from the others, with values of (θs)1 varying rapidly close to the surface and in the entrainment region. Indeed, the ASTEX vertical profiles presented in Cuijpers and Bechtold (1995) correspond to a moist surface layer with a dryer and colder PBL than the other FIRE-I, EPIC or DYCOMS-II observed vertical profiles. This kind of diagram can illustrate to what extent a vertical profile may be typical of a characteristic stratocumulus pattern.

11. Thermodynamic diagrams

As stated by Emanuel (1994, chapter 5), ‘the stability characteristics and thermodynamic properties of convective clouds and of convecting atmospheres are most easily seen by making plots of the thermodynamic variables. Various thermodynamic transformations can also be easily calculated using thermodynamic diagrams, avoiding the often tedious calculations necessary in moist thermodynamics’.

Accordingly, it is possible to add a new set of moist entropy curves (based on (θs)1) on the so-called skew T–logp diagram, as a companion set of the dry entropy curves (dry convection/θ) and of the pseudo-potential temperature curves (deep convection/θw).

Figure 14 is an example of a skew T–logp diagram where an initial parcel defined by p = 1000 hPa, T = 20°C and qv = 4 g kg−1 is shifted upwards adiabatically to 250 hPa, with the assumption of a constant value for the moist entropy (surface value of θs = 27°C). The moist entropy temperature Ts (open circle) is defined for each level as the value of θs measured at the corresponding condensation level, in a way similar to the graphical process used to evaluate θw.

Figure 14.

The skew T–logp diagram. The classic contours of T, qv, θ and θw are depicted in the usual way. The moist entropy solid lines are defined by constant values of θs and they are labelled by boxed values from −20 to 120 °C. They tend toward the corresponding dry adiabatic values θ for small values of qv (upper left) and the differences increase for larger values of qv (bottom right). An ideal and adiabatic ascent of a parcel is depicted by dark circles for Td, open circles for Ts and dark squares for T. (Further explanations are given in the text.)

For this ideal case-study and above the condensation level, the θs = 27°C line is located between the unsaturated dry adiabatic line (θ = 20°C) and the saturated pseudo-adiabatic one (θw = 10°C). It can be noted that, above the condensation level, liquid or ice cloud water exist and are taken into account in the computations of θs.

For real non-precipitating ascents (such as shallow convection), diabatic processes exist (horizontal or vertical advections, radiation, lateral mixing with the environment), and they all modify the ascent in a way to be determined for each case.

An expansion of the skew T–logp diagram is presented in Figure 15, where the vertical profile of the FIRE-I RF03B dataset is plotted up to 700 hPa. The PBL top is at 904 hPa for that flight. The moist entropy temperature Ts (open circle) corresponds to the value of (θs)1 computed at each level from the data flight and taking into account the cloud liquid water.

Figure 15.

An expansion of the skew T–logp diagram for the FIRE-I RF03B stratocumulus. The dark circles (on the left) represent the dew point temperatures Td, the open circle (middle position) the moist entropy temperatures Ts, and the dark squares (on the right) the dry-bulb temperatures T. The moist PBL is characterized by constant values for Ts up to the level 904 hPa, including the entrainment region which extends up to the first level where qv < 3 g kg−1 (Figure 1(a)). Further explanations are given in the text.

The PBL is characterized by an almost constant value of Ts, remaining close to 304.5 K (or 31.5 °C) for both the saturated and the unsaturated layers, as already suggested in Figure 3. The two lines θs = 30 and 32 °C are depicted, in order to make the analysis easier.

It can be noted that, up to the surface condensation level (about 960 hPa), the RF03B ascent looks like the ideal ascent depicted in Figure 14, with a saturated constant Ts path up to the PBL top (904 hPa). Above the PBL top, the jump in Ts is small (perhaps less than 1 C) and the moist entropy temperature Ts increases linearly with z or logp in the dry and warm subsiding air, due to the diabatic processes (radiation and subsidence).

12. The budget equation for moist entropy

Although it is a central question in this article, it may be difficult to understand or to explain why moist entropy seems to be almost a constant throughout the PBL region of marine stratocumulus, as observed in section 7.1. The difficulty lies in the Second Law of Thermodynamics, which is difficult to apply to real atmospheric circulations, particularly if stationary fluxes of heat and water species exist at the surface, and are transmitted by conduction, turbulence or convective processes to higher atmospheric levels.

One of the ways to understand ‘by hand’ why the profile of moist entropy may be a constant within the PBL of marine stratocumulus clouds is to analyze the properties displayed by these clouds in the atmosphere, and by entropy in general thermodynamics.

  • (Atmosphere) In marine stratocumulus, it is assumed that the cloud and the sub-cloud regions are in quasi-equilibrium with the surface temperature and the thermal radiation. This kind of cloud acts as a ‘black body’ radiator. Even if sources and sinks of energy and species exist at the surface and at the top of the cloud, it is an open system in a quasi-equilibrium and a quasi-stationary state.

  • (Thermodynamics) In contrast to a closed system, steady states with constant entropy production are possible for open systems. If the system is sufficiently close to equilibrium, the local equilibrium hypothesis can be made and, from the Prigogine theorem, the entropy production is extremal, with a constant entropy production balanced by removal from the system, so that the entropy may be locally held constant.

  • (Turbulence) Since the moist turbulent processes act in order to mix up the steady-state properties with no sources or sinks, and since moist entropy has indeed no (or small) sources or sinks within the PBL of marine stratocumulus, moist entropy must be well-mixed throughout the PBL (the MIME process), contrary to the Betts variables which must vary with height in order to be in equilibrium with the steady-state vertical fluxes of energy and water species.

Another way to try to understand why moist entropy may be a constant is the analysis of the material change for moist entropy. From (C.1) and (C.5), the following statements are confirmed:

equation image(75)

If the marine stratocumulus clouds are in a quasi-equilibrium and quasi-stationary state, with a net energy flux due to radiation almost equal to zero inside the cloud, or somehow balanced with other sources/sinks, the net value equation image may be considered as a small term in (75). It is also assumed that, except close to the surface, the dissipation term equation image is a small term. As demonstrated in Appendix C, the bracketed term in (75) represents the condensation and evaporation processes and it is cancelled out for a set of reversible changes of phase. As a consequence, the first line on the RHS of (75) is almost equal to zero for a marine stratocumulus and for the reversible and moist adiabatic cycle represented in Figure 16(a). In that case, the budget equation for moist entropy is controlled by the two other terms in the second line of (75), which both depend on the diffusion fluxes Jk for dry air and water species. If no precipitation exists and if no external mixing occurs between the different species of the moist air, then the diffusion fluxes are small or equal to zero, leading to ds/dt = 0 and to a possible explanation for the conservative property displayed by the moist entropy within the PBL region of marine stratocumulus.

Figure 16.

Schematic representations of three Lagrangian motions occurring inside (or close to) a stratocumulus region: (a) a reversible and moist adiabatic cycle, (b) a PBL-top entrainment of a clear-air parcel within the stratocumulus, and (c) lateral mixing or exchanges between the stratocumulus air (moist or dry) and the warmer cloud-free environment (or for the transition to a cumulus case).

The process represented in Figure 16(b) corresponds to an entrainment of a warm and dry clear-air parcel through the top of the stratocumulus. When the parcel enters the cloud, the solar radiation is gradually switched off and equation image becomes a small term in (75). The entrainment is then associated with a cooling of the parcel, a saturation toward esw and a condensation of liquid water. The cooling occurring after the entrainment may be explained by a thermal equilibrium process between the warm parcel and the colder surrounding cloud air. The reason why the temperature is lower inside the cloud cannot be explained by the entropy budget. It corresponds to the First Law and the internal energy or the enthalpy budgets. The saturation and the condensation processes undergone by the parcel are associated with almost reversible changes of phase, leading to a cancellation of the bracketed term. Therefore the three terms in the first line on the RHS of (75) are small. If the diffusion fluxes Jk are assumed to be small, then the entropy and (θs)1 must be conservative quantities, with the PBL-top values retained within the cloud, after the entrainment stage.

At the edges of the cloud (or outside the clouds, for cumulus cases), the net heating rate due to radiation (equation image) is not equal to zero, leading to higher values close to the surface for the moist entropy and with (θs)1 decreasing with height, as depicted in Figure 16(c). The exchanges between the stratocumulus and the lateral cloud-free air may gradually modify the moist entropy of the stratocumulus (and vice versa). The lateral cloud-free vertical profile for (θs)1 corresponds to a composite analysis (not shown), realized by the author for several shallow cumulus cases (BOMEX, ARM-Cu, RICO-composite, ATEX, GATE, SCMS-RF12).

Another way to understand how the existing jumps in θl and qt can be in agreement with a continuous profile of the moist entropy and of θs at the top of the PBL is presented in Figure 17. Following the graphical approach of Gibbs (1873a,b), a 3D curve θs(θl,qt,z) is plotted in Figure 17(a), with the Betts variables as horizontal coordinates and with the usual Betts vertical profiles obtained by projections onto the left and the rear vertical planes, where large jumps exist for θl and qt. The ‘mystery’ of the disappearing jump in θs is explained in Figure 17(b), by a view ‘in profile’ of the 3D curve of θs(z) when it is projected onto the slantwise plane normal to the vertical isentropic planes.

Figure 17.

A 3D representation of the curve (θs)1(qtl,z) (heavy black line). The conserved variable diagram is placed at the bottom, with a schematic curve representing a typical behaviour of the curves depicted in Figure 13, with the same slantwise grey solid lines corresponding to constant values for (θs)1 (moist isentropes). (a) The 3D curve is obtained by plotting for each height z the point of coordinates (qtl), with the vertical light grey arrows connecting the light grey conserved variable curve to the heavy black 3D curve. The curves qt(z) and θl(z) are obtained from the 3D curve by projections onto the left and the rear planes, respectively, with large jumps observed not only for qt(z), θl(z) but also for the 3D curve. (b) The new curve (θs)1(z) is obtained by a projection onto the slantwise vertical plane normal to the moist isentropic vertical plane. Even if the jumps in qt(z) and θl(z) are large, the jump in (θs)1(z) almost disappears because the 3D curve is almost parallel to the iso-(θs)1(z) vertical plane, leading to a straight line (up to the top of the inversion) created by the projection onto the plane normal to it. The 3D curve starts to diverge from the mean iso-(θs)1(z) plane above the top of the inversion and, accordingly, the curve (θs)1(z) starts to increase in the clear air above the stratocumulus. The direction of increasing potential temperature θX(θl,qt) is depicted in the conserved variable diagram (at the bottom) as a normal to the gradient in (θs)1(θl,qt).

The jumps in θl(z) and qt(z) are thus minimized in the direction normal to the isentropic plane, labelled by (θs)1, whereas they are maximized in the direction parallel to the isentropic plane, labelled in Figure 17(b) by the normal coordinate denoted by θX and defined from (63) by

equation image(76)
equation image(77)

A possible application of these normal variables (θs)1 and θX are the vertical flux of them, approximated by (64) and (65) for (θs)1 and by

equation image(78)

for the vertical flux of θX. It is possible to invert (64), (65) and (78) to express the fluxes of the Betts variables as

equation image(79)
equation image(80)

The system (79) and (80) corresponds to the local relations

equation image(81)
equation image(82)

The aim of the flux of (θs)1 is to reduce the departures from an isentropic profile, whatever the flux of θX may be. The aim of the flux of θX is to jointly reduce the vertical gradients in θl and qt, under the constraint of a conserved moist entropy. This system (79) and (80) may lead to new analyses or modelling of the moist turbulent processes.

13. Conclusions

It is demonstrated in this article that the moist potential temperature θs is a true synonym of moist entropy, whatever the standard and reference values T0, Tr or pr may be. It is suggested that θs could be an answer to the questions raised in the introduction of HH87: it ‘can be regarded as a direct measure of (moist) entropy’, it ‘stresses the importance of (moist) entropy in atmospheric dynamics’, and it could suggest some hints on ‘how entropy can be used in cloud modelling’.

The analysis of the FIRE-I data flights shows that stratocumulus exhibits an almost constant moist entropy regime within the whole PBL (from the surface to the top of the cloud). Moreover, it seems that there is no (or only a small) jump in moist entropy at the top of the stratocumulus, with a soft and continuous transition between the moist PBL and the warm and subsiding dry air above. The explanations for these observed features are still partly unclear, although it has been explained via a 3D perspective why it is possible to have at the same time large jumps in θl and qt and a smooth profile for moist entropy.

It is shown that moist entropy can be approximated by a simple expression denoted by (θs)1 and given by any of (45), (46), (47), (49) or (50), with a good accuracy and with the common values Λ = 5.87 valid for all flights. It can be noted that all these formulae can be applied to either liquid water or ice cloud drops. Therefore, they can be applied in GCMs or LAMs, including over polar regions.

The comparison of θs and (θs)1 with the well-known Betts (1973) liquid-water potential temperature θl shows that an extra term Λ qt appears, with the coefficient Λ corresponding to the difference between the dry air and the water vapour partial entropies. It is a way to take into account the impact of the change in entropy when some dry air enters a parcel of fluid and when it is replaced by water vapour, and vice versa. These kinds of processes were not fully represented in any of the previous potential temperature computations.

The mixing in moist entropy process (MIME) appears to correspond to the CTEI criterion curves suggested by Randall (1980) and Deardorff (1980). The slantwise lines representing constant values for (θs)1 can be used in conserved variable diagrams to represent the stratocumulus curves. It is also possible to represent the moist entropy lines–or iso-(θs)1 curves–in the skew T–logp diagrams, with clear distinctive patterns valid for marine stratocumulus clouds, as observed in many real soundings (not shown).

Since moist entropy and the corresponding moist potential temperature (θs)1 are constants within the moist PBL in all FIRE-I data flights, and also for the ASTEX, EPIC and DYCOMS-II (RF01) cases, it may be interesting to use (θs)1 to study the non-precipitating stratocumulus. The applications may also concern the more general case of non-adiabatic turbulent fluxes, with the Betts variable fluxes expressed in (79) and (80) in terms of two weighted sums of the turbulent fluxes of (θs)1 and θX. This formulation offers new perspectives, with the flux of (θs)1 acting as a relaxation term toward a constant vertical profile of entropy, whereas the flux of θX may act as an isentropic and joint mixing of θl and qt. It can be noted that the problem of reprojection onto the non-conservative variables is not approached in this study.

It may be interesting to express the flux of θv in the thermal production (involved in the prognostic turbulent kinetic energy equations) in terms of the fluxes of (θs)1 and maybe θX, with possible large impacts for both saturated and unsaturated moist air.


The author is most grateful to J.-F. Geleyn, J. L. Brenguier, P. Santurette, I. Sandu and J. M. Piriou for helpful suggestions and encouraging discussions. The author would like to thank the anonymous referees for their constructive comments, which helped to improve the manuscript.

The validation data from the NASA flights during the FIRE I experiment were kindly provided by S. R. de Roode and Q. Wang.

Appendix A.

List of acronyms and symbols

ARMAtmospheric Radiation Experiment
ARPEGEAction de Recherche Petite Echelle et
 Grande Echelle
ASTEXAtlantic Stratocumulus Transition Experiment
ATEXAtlantic Tradewind Experiment
BOMEXBarbados Oceanographic and Meteorological
CALIPSOCloud Aerosol Lidar and Infrared Pathfinder
 Satellite Observations
CRMCloud-resolving model
CTEICloud-Top Entrainment Instability
DYCOMSDynamics and Chemistry
 Of Marine Stratocumulus
EPICEast Pacific Investigation of Climate
EUCLIPSEEuropean Union CLoud Intercomparison,
 Process Study and Evaluation project
EUROCSEUROpean Cloud Systems
FIREFirst ISCCP Regional Experiment
GARPGlobal Atmospheric Research Program
GATEGARP Atlantic Tropical Experiment
GCMGeneral circulation model
GCSSGEWEX Cloud System Study
GEWEXGlobal Energy and Water cycle Experiment
IFSIntegrated forecasting system
IPCCIntergovernmental Panel on Climate Change
ISCCPInternational Satellite Cloud Climatology
LAMLimited-area model
LESLarge-eddy simulation
MIMEMixing In Moist Entropy
NWPNumerical weather prediction
PBLPlanetary boundary layer
RICORain in Cumulus over the Ocean
SCMSingle-column model
SCMSSmall Cumulus Microphysics Study
TRMMTropical Rainfall Measuring Mission
equation imageHorizontal and linear averaging operator
<… >Horizontal and logarithmic averaging operator
cpdSpecific heat for dry air (1004.7 J K−1kg−1)
cpvSpecific heat for water vapour
 (1846.1 J K−1kg−1)
clSpecific heat for liquid water (4218 J K−1kg−1)
ciSpecific heat for ice (2106 J K−1kg−1)
cpSpecific heat at const. pressure for moist air
 = qdcpd + qvcpv + qlcl + qici
 = qd (cpd + rvcpv + rlcl + rici)
c= cpd + rtcl
equation image= cpd + rtcpv
d/dtMaterial (Lagrangian) barycentric derivative
equation imageKinetic energy dissipation rate
eWater vapour partial pressure
erWater vapour reference partial pressure
 = esw(T0) ≈ 6.11 hPa
esw(T)Partial saturating pressure over liquid water
esi(T)Partial saturating pressure over ice
hSpecific enthalpy
hdSpecific enthalpy for dry air
hvSpecific enthalpy for water vapour
hlSpecific enthalpy for liquid water
hiSpecific enthalpy for ice water
Lvap(T)= hvhl, latent heat of vaporisation
Lsub(T)= hvhi, latent heat of sublimation
Lfus(T)= hlhi, latent heat of fusion
Lvap(T0)= 2.501 × 106 J kg−1
Lsub(T0)= 2.835 × 106 J kg−1
Lfus(T0)= 0.334 × 106 J kg−1
p= pd + e, local value for the pressure
pr= (pd)r + er, reference pressure (pr = p0)
pdLocal dry air partial pressure
(pd)rReference dry air partial pressure (≡ prer)
p0= 1000 hPa, conventional pressure
qd= ρd, specific content for dry air
qv= ρv, specific content for water vapour
ql= ρl, specific content for liquid water
qi= ρi, specific content for ice water
qt= qv + ql + qi, total specific content for water
qrReference specific content of water
 = rr/(1 + rr) ≈ 3.84 g kg−1
equation imageRate of change of ql into qv (evaporation)
equation imageRate of change of qi into qv (sublimation)
equation imageRate of change of qi into ql (fusion)
qswSaturation specific content for water vapour
equation imageEffective diabatic heating rate
rv= qv/qd, mixing ratio for water vapour
rl= ql/qd, mixing ratio for liquid water
ri= qi/qd, mixing ratio for ice
rt= qt/qd, mixing ratio for total water
rrReference mixing ratio for water species,
 with ηrrer/(pd)r, → rr≈ 3.82 g kg−1
rSwSaturation mixing ratio for water vapour
R=qdRd + qvRv, gas constant for moist air
 =qd (Rd + rvRv)
RdDry air gas constant (287.06 J K−1kg−1)
RvWater vapour gas constant
 (461.53 J K−1kg−1)
R=Rd + rtRv
SmMoist entropy static energy
SlLiquid-water static energy
sSpecific entropy
sdSpecific entropy for dry air
svSpecific entropy for water vapour
slSpecific entropy for liquid water
siSpecific entropy for ice
equation imageApprox. specific entropy for dry air
equation imageApprox. specific entropy for water vapour
equation imageApprox. specific entropy for liquid water
srReference entropy
(sd)rReference values for entropy of dry air,
 at T0 and (pd)r
(sv)rReference values for entropy of
 water vapour, at T0 and er
equation imageStandard specific entropy for dry air
 (value at T0 and p0 : 6775 J K−1kg−1)
equation imageStandard specific entropy for water vapour
 (value at T0 and p0 : 10320 J K−1kg−1)
equation imageStandard specific entropy for liquid water
 (value at T0 and p0 : 3517 J K−1kg−1)
equation imageStandard specific entropy for ice
 (value at T0 and p0 : 2296 J K−1kg−1)
TLocal temperature
TdDew point temperature
TrReference temperature (TrT0)
TsMoist entropy temp. corresponding to θs
T0Zero Celsius temperature (= 273.15 K)
α= 1, specific volume
γ= η κ = Rv/cpd ≈ 0.46
δ= Rv/Rd − 1 ≈ 0.608
δkj= 1 if k = j; = 0 otherwise
ε= 1 = Rd/Rv ≈ 0.622
η= 1 + δ = Rv/Rd ≈ 1.608
θ= T (p0/p)κ, potential temperature
θwWet-bulb pseudo-adiabatic potential temp.
θEEquivalent potential temp.
θESSaturation equivalent potential temp.
θvVirtual potential temp. (L68)
θlLiquid-water potential temp. (B73)
θilIce–liquid water potential temp. (TC81)
θvlLiquid-water virtual potential temp. (GB01)
equation imageLiquid-water virtual potential temp. (E94)
equation imageEntropy temperature (HH87)
θMoist entropy potential temp. (M93)
equation imageReference value for θ (M93)
θsNew moist entropy potential temp.
(θs)1Approximate version of θs (1st part)
(θs)2Approximate version of θs (2nd part)
θsrReference value for θs
θXCoordinate normal to θs.
κ= Rd/cpd ≈ 0.2857
λ= cpv/cpd − 1 ≈ 0.8375
Λ= {(sv)r − (sd)r}/cpd ≈ 5.87
μk= hkTsk, specific chemical potential
 for the species k =(d, v, l, i)
Π= T/θ, the Exner function
ρSpecific mass for moist air
 = ρd + ρv + ρl + ρi
ρdSpecific mass for dry air
ρvSpecific mass for water vapour
ρlSpecific mass for liquid water
ρiSpecific mass for ice water
ω= dp/dt, vertical wind in isobaric coords.

Appendix B.

The moist potential temperatureθs

The specific moist entropy is defined by (B.1) as a weighted sum of the specific partial entropies and, following HH87, it can be expressed as (B.2), where qt = qv + ql + qi.

equation image(B.1)
equation image(B.2)

The differences of the partial entropies can be expressed in terms of the differences of the enthalpies and the chemical potentials, leading to

equation image(B.3)
equation image(B.4)

The differences of the enthalpies are equal to the latent heats Lvap = hvhl and Lsub = hvhi. If metastable states such as supercooled water are ignored, the difference of the chemical potentials are equal to the affinities and they are related to the saturation partial pressures by

equation image(B.5)
equation image(B.6)

When (B.3)–(B.6) are inserted into (B.2), they yield

equation image(B.7)

The second line of (B.7) cancels out for clear-air regions, where ql = qi = 0. It is also equal to zero for cloudy air if the partial pressure of the water vapour is equal to esw if ql0, or is equal to esi if qi0 (i.e. with no un- or supersaturation).

For the atmospheric conditions where the specific heat and the gas constants do not vary with T or p, the dry air and water vapour specific partial entropies sd and sv can be expressed analytically as a relative change from a given reference state, defined by Tr, (pd)r, er and (qv)r = qr:

equation image(B.8)
equation image(B.9)

When (B.8) and (B.9) are inserted into (B.7), with the second line cancelled, they yield

equation image(B.10)

The M93 formulation of the quotient equation image follows from (B.10) and from the definition (24) in section 3, with a rearrangement of the terms expressed as qdcln().

The computation of the quotient θssr defined by (39) in section 4 is obtained by transforming qd (sd)r + qt (sv)r in (B.10) with the property qd = 1 − qt, leading to

equation image(B.11)

where Λ = {(sv)r − (sd)r}/cpd. Similarly,

equation image(B.12)

where λ = ( cpvcpd )/cpd.

A reference value qr is introduced in (B.11), with the use of a logarithm, to give

equation image(B.13)

The next step is to insert (27), (28) and (B.13) into (B.10), together with the following relations defined for the reference state:

equation image(B.14)
equation image(B.15)
equation image(B.16)
equation image(B.17)

After some rearrangement of the terms, the result is written with all the varying terms expressed as cpd ln(), leading to

equation image(B.18)

The quotient θssr and the formulations (40) and (41) for θs and θsr follow directly from the identification of all the logarithm terms in (B.18) with the one in (39).

Appendix C.

The conservative equation for (θs)1

The formalism used in this Appendix is adapted from the approaches of De Groot and Mazur (1962), M93 or Zdunkowski and Bott (2004). The implicit Einstein's summation rules prevail with k = 0,1,2,3 representing the dry air, the water vapour, the condensed liquid water, and ice, respectively. The material derivative d/dt for any variable can be separated into a sum of external and internal changes dex/dt + din/dt.

The external changes dex/dt are generated by the diffusion fluxes of matter Jk, with the differential velocity computed for each component with respect to the barycentric mean velocity v, leading to Jk = δkjρj (vjv). The external changes of matter dex(qk)/dt are equal to −(ρ)−1 · Jk. The internal changes din/dt are generated by the physical processes such as the absorption of radiation or the phase changes, regarded as chemical reactions.

The effective diabatic heating rate equation image will be defined as the sum of the true internal diabatic heating rate (equation image) plus the kinetic energy dissipation, equation image, plus the differential diffusion of the partial enthalpy, hk, leading to

equation image(C.1)

It can be noted that the latent heat release processes are not included in equation image (nor in equation image). They are represented by the internal changes din(qk)/dt.

With the use of (C.1), the enthalpy and the entropy equations are given by

equation image(C.2)
equation image(C.3)

The entropy equation (C.3) is equivalent to the Gibbs equation (1), with the material derivatives replacing the differentials.

The derivative of h = qkhk is equal to qkdhk/dt + hkdqk/dt. The two terms are equal to qkcpkdT/dt = cpdT/dt and hkdexqk/dt + hkdinqk/dt, respectively. For hydrostatic equilibrium, (ρ)−1 = RT/p, resulting in the temperature and the entropy equations being written

equation image(C.4)
equation image(C.5)

The bracketed term in (C.5) can be evaluated for a set of adiabatic internal changes given by

equation image(C.6)
equation image(C.7)
equation image(C.8)

They represent the conversions between the water species, as in section 2.1 for the Betts approach, except that all the conversion terms are included, i.e. with evaporation (or condensation), sublimation (or solid condensation) and fusion (or solidification) processes. The latent heat release processes are represented by (C.6)–(C.8), with the corresponding impacts −hkdinqk/dt and −μkdinqk/dt in the enthalpy and entropy equations, respectively.

From (C.6)–(C.8), the bracketed term in (C.5) is written

equation image(C.9)

These terms vanish if changes of phase are assumed to be reversible and to occur with zero affinities, i.e. with the same chemical potentials μk. It is true if no supersaturation nor metastable phases exist (such as liquid water with T < T0).

The aim of this section is to verify that (C.5) is almost valid for the moist entropy s defined by (39) and with θs approximated by (θs)1 given by (46). Also, it would be important to understand how the approximate entropy equation defined with (θs)1 works with open systems and variable values for qd and qt. The resulting equation, valid for cpd ln[(θs)1] can be written

equation image(C.10)


equation image(C.11)

Let us assume the following hypotheses:

equation image(C.12)
equation image(C.13)
equation image(C.14)

When (C.4) is put into (C.14), and then into (C.10) via (C.11), the approximate equation results:

equation image(C.15)

The approximate formula (C.15) has been obtained with the last term in (C.4) transformed into

equation image

and with cpd Λ = (sv)r − (sd)r and (C.12) plus (C.13) introduced into (C.10).

All the terms in the second and third lines of (C.15) do not exist in (C.5). Therefore, the challenge is to understand under what conditions these terms can vanish in open systems, where not only reversible exchanges can exist between the water species qv, ql and qi, but where qd and qt can also vary, with however the conservative constraint dqd/dt = −dqt/dt.

The next step is to write the identities

equation image(C.16)
equation image(C.17)


equation image(C.18)

With (C.16)–(C.18), the second and third lines of (C.15) are changed into

equation image(C.19)

The two terms in the second line of (C.19) depend on differences in chemical potentials. They must be evaluated for both external and internal changes in qk. For the external changes, the chemical potentials are written with (B.5) and (B.6) and for the internal changes the set of internal conversions (C.6)–(C.8) are put into (C.19), leading to

equation image(C.20)

The last three terms of (C.20) exactly cancel out if the change of phases are reversible ones, i.e. if the chemical potentials are equal if one of the corresponding conversion rates equation image, equation image or equation image exists.

The second line of (C.15) does not exactly cancel out. Nevertheless, it can be assumed that, if some liquid water enters or leaves the parcel via the external diffusion fluxes (i.e. due to departures from the mean barycentric motion), the partial pressure e for the water vapour will be equal to its saturating value esw, in order to deal with isentropic and reversible processes. The same is true for isentropic and reversible changes in the ice water, for which it is assumed that e = esi if some qi enters or leaves the parcel.

Similarly, the first line of (C.20) does not cancel out, since svsd is not exactly equal to (sv)r − (sd)r. However, it is expected that the difference (svsd) − cpd Λ must be much smaller than (svsd), leading to larger errors if the terms cpd Λ were omitted in (C.20), as in the Betts formulation θl. If this term was not included, a diffusion of qt into qd (or vice versa) would lead to an impact much more important than with (θs)1 defined by (46) and leading to the first line of (C.20).

To confirm these statements, let us write the difference (svsd) − [(sv)r − (sd)r] as

equation image(C.21)

As for the difference (svsd), it can be evaluated with equation image and equation image as absolute reference values, leading to

equation image(C.22)

For the values Tr = T0 and (pd)rp0 retained in the present study, the difference of (C.22) from (C.21) is equal to the last bracketed terms of (C.22).

For the values of the constants given in the Appendix A, this difference can be evaluated to −2352 − 3545 = −5897 J K−1kg−1. The other terms of (C.21) are equal to zero for T = Tr, pd = (pd)r or e = er. For the extreme tropospheric values T = 320 K, pd = 50 hPa or e = 0.1 hPa, the three terms of (C.21) are equal to +134, +860 and −1896 J K−1kg−1, respectively. Therefore, the magnitudes of the first two terms depending on ln(T/Tr) and ln(pd/(pd)r) are indeed small in comparison to 5897 J K−1kg−1. The last term depending on ln(e/er) is less than one third of 5897 J K−1kg−1 for e = 0.1 hPa (upper troposphere values). For the FIRE-I region, qv varies between 2 and 10 g kg−1 for p = 850 and 1000 hPa, leading to values of e varying between 3 and 16 hPa, with the last term Rv ln(e/er) varying between 328 and 444 J K−1kg−1. It is thus less than one tenth of 5897 J K−1kg−1.

As a consequence, the explanation of how the approximate entropy equation (C.15) works with open systems and with variable values for qd and qt highlights the importance of the term Λqt in the formulation of θs or (θs)1, and in (C.10).

Appendix D.

The averaging operators

Conditionally linear averages can be applied to the specific contents qv, ql, qi or qt = 1 − qd. However, they must not be applied to θl or θs, because only the moist entropy s displays an additive property, with the moist entropy depending on cpd times the logarithm of θs and with equation image.

Accordingly, the ‘logarithmic mean value’ for θs will be denoted by s>. It is valid for either the clear-air, the in-cloud or the grid-cell averages of the entropy equation image. It is defined by averaging (39) with qr, (sd)r, (sv)r, cpd and θsr constant, leading to

equation image(D.1)


equation image(D.2)

Consequently, the logarithmic mean of < (θs)1 > is defined by (45)–(47), leading to the result

equation image(D.3)

The nonlinearity concerns the logarithm term and the joint variations of T and ql or qi in the last terms of (D.3).

For the FIRE-I flights, the local values of (θs)1 mainly vary on the horizontal and they remain close to the mean value equation image with a discrepancy of a few percent. In such a case, the departure term equation image is smaller than the average value equation image and the term

equation image

can be approximated with ln(x) ≈ xx2/2 by equation image, leading to

equation image(D.4)

For horizontal fluctuations of (θs)1, the departure term equation image can be discarded because, for |(θs)′1| less than 5 K and for equation image equal to 300 K, the departure term is about 3 × 10−4, leading to an impact of 0.05 K on <(θs)1>.

As a consequence, the horizontal mean value for (θs)1 is written

equation image(D.5)

The same analysis could not be retained for an application to a vertical mean of the moist entropy, with possible larger departure terms |(θs)′1| (i.e. for an averaging of the PBL values and the free upper-air regions). In that case, the formulae (D.3) or (D.4) must be retained.

For the specific contents qv, ql, qi or qd = 1 − qt, the standard deviation σq is obtained from the linear mean value equation image and the corresponding variance equation image, leading to the result

equation image(D.6)

The method is different for a moist potential temperature like s> defined by (D.3). If the mean and the variance of ln(θ) are denoted by equation image and equation image, the standard deviation for ln(θ) is given by (D.7) and ln(θ) can vary within mln(θ) ± σln(θ). The standard deviation for θ can be set to half of the spread width exp[ mln(θ) ± σln(θ)], leading to the results expressed by the product (D.8), valid for the potential temperature θ.

equation image(D.7)
equation image(D.8)