On a general definition of the squared Brunt–Väisälä frequency associated with the specific moist entropy potential temperature

Authors


Abstract

The squared Brunt–Väisälä frequency (BVF) is computed in terms of the moist entropy potential temperature recently defined by Marquet (2011. Q. J. R. Meteorol. Soc. 137: 768–791). Both homogeneously saturated and non-saturated versions of N2 (the squared BVF) are derived. The method employed for computing these special homogeneous cases relies on the expression of density written as a function of pressure, total water content and specific moist entropy only. The associated conservative variable diagrams are discussed and compared with existing ones. Despite being obtained without any simplification, the formulations for N2 remain nicely compact and are clearly linked with the squared BVF expressed in terms of the adiabatic non-saturated and saturated lapse rates. As in previous similar expressions, the extreme homogeneous solutions for N2 are of course different, but they are not analytically discontinuous. This allows us to define a simple bridging expression for a single general shape of N2, depending only on the basic mean atmospheric quantities and on a transition parameter, to be defined (or parametrized) in connection with the type of application sought. This integrated result remains a linear combination (with complex but purely local weights) of two terms only, namely the environmental gradient of the moist entropy potential temperature and the environmental gradient of the total water content. Simplified versions of the various equations are also proposed for the case in which the moist entropy potential temperature is approximated by a function of both so-called moist conservative variables of Betts. Copyright © 2012 Royal Meteorological Society

1. Introduction

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

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

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

This article is organized as follows. The mathematical definition of the moist squared BVF (equation image) is presented in section 2, with equation image expressed in terms of the gradients of the two conservative variables (s,qt), and is established in Appendix B. The moist definition of the state equation is recalled in section 3, together with M11's specific moist entropy defined in terms of θs. The non-saturated and saturated versions equation image and equation image are then presented in sections 4 and 5, with some detailed computations available in Appendices C and D, respectively.

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

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

2. The moist squared Brunt–Väisälä frequency

It is shown in Appendix B that the moist squared BVF can be defined by

equation image(1)

Formulation (1) is different from the classical one used in DK82 or E94, for instance. It is assumed that the density can be expressed as a function of the two conserved variables s and qt, as well as of pressure p, leading to ρ = ρ(s,qt,p). The moist squared BVF can then be expressed as a weighting sum of the two local vertical gradients of s and qt, with weighting factors depending on appropriate partial derivatives of the density with respect to s and qt.

In order to compute the moist formulation (1), the density must be expressed analytically in terms of the three independent variables (s,qt,p). One of the problems is that such an explicit formulation for ρ(s,qt,p) does not exist for the moist air. It is, however, possible to define ρ in terms of (s,qt,p) by expressing the entropy and the state equations as ρ(T,qt,p) and s(T,qt,p), and then by eliminating the temperature. This method is used in sections 4 and 5 and in Appendices C and D to compute the non-saturated and saturated versions of the squared BVF.

The saturated case is more difficult to deal with than the non-saturated one, because of the existing condensed water terms ql or qi. However, these terms can be expressed as differences between qt and the saturated values qsw or qsi, which depend on T, p and qt. It is thus necessary to express the temperature T in terms of (s,qt,p), at least implicitly.

3. The state and the entropy equations

The state equation of the moist air is written with the use of the dry gas constant Rd replacing the moist value R = qd Rd + qv Rv, yielding

equation image(2)

where the virtual temperature Tv (Lilly, 1968) is equal to

equation image(3)
equation image(4)

From Appendix A, the constants are equal to δ = Rv/Rd − 1 ≈ 0.608 and η = δ + 1. The virtual temperature corresponds to Eq. (9) in DK82 and to the ‘density temperature’ denoted by Tρ in E94. The associated (liquid water) virtual potential temperature is equal to θv defined in (3).

The specific moist entropy is defined in M11 as

equation image(5)

where the reference entropy sref is equal to

equation image(6)

and where the entropy potential temperature θs can be written as

equation image(7)

The advantage of definition (5) for s, with cpd and sref being two numerical constants, is that θs becomes truly synonymous with the specific moist entropy, whatever the thermodynamic properties of the parcel (i.e. T, p, ..., qt, ql, rv, ...) may be.

It can be verified (see M11) that both sref and θs are independent of the reference values Tr and pr, providing that rr(Tr,pr) is equal to the saturating values ε / [ pr/esw(Tr) − 1 ] or ε / [ pr/esi(Tr) − 1 ], depending on Tr > T0 or Tr < T0, respectively. All the different terms of sref and θs depend on Tr and pr in such a way that sref and θs remain unchanged.

The new term Λr appearing in (7) is the main difference from previous studies on moist entropy and associated moist potential temperatures. It depends on the difference between the reference partial entropy values of dry air and water vapour, leading to the numerical value Λr = [(sv)r − (sd)r]/cpd ≈ 5.87 (as computed in M11 from the Third Law and for the reference state, see Appendix A for the reference values of entropies).

The impact of the reference values of partial entropies on the definition of the moist air entropy has been addressed independently in Pauluis et al. (2010, hereafter referred to as PCK10), where an equivalent of Λr has been studied in the Appendices, in the form of an unknown and arbitrary constant ‘a’.

In contrast, the moist entropy is computed in E94 and P11 ‘per unit mass of dry air’, with the reference values (sd)r and (sv)r suppressed from the formulae of the specific values sd and sv. This method prevents the possibility of arriving at a relevant definition for the specific moist entropy (‘per unit mass of moist air’), with varying values for qd or qt = 1 − qd to be put in factors of (sd)r and (sv)r, respectively.

The three terms on the right-hand side of the first line of (7) correspond to the variables θl and qt defined in Betts (1973, hereafter referred to as B73), with qt multiplied by Λr. The last term on the second line of (7) might be irrelevant for the dry-air limit where qt = qv and rv = qv/(1 − qv) tends to 0 when qv tends to 0. However, the term varies like qv ln(qv), which has the limit 0 when qv tends to 0.

4. The squared BVF for unsaturated moist air

The unsaturated moist air is defined by ql = qi = 0 and qt = qv. From (5) and since cpd is a constant, the gradient of the specific moist entropy is exactly equal to

equation image(8)

The computations of the two partial derivatives of the density involved in the formulation (1) for equation image are described in Appendix C. They are given by (C6) and (C13), leading to

equation image(9)

The term Λv and the unsaturated adiabatic lapse rate Γns are equal to

equation image(10)
equation image(11)

The derivative of T with respect to pressure is computed in (11) at constant specific moist entropy and water content. The resulting value Γns, with cp attaining its moist value depending on qv, has been obtained after long computations involving all terms entering into the specific moist entropy formulation defined by (5)–(7), in a way analogous to the computations described in some detail in Appendix C.

The term in (9) involving the gradient of qd can be written in any of the alternative ways

equation image(12)
equation image(13)

The formulation involving qd corresponds to the original one derived in Lalas and Einaudi (1974, hereafter referred to as LE74) for the saturated case. It is retained for the present non-saturated version in order to be consistent with the next section.

It is important to notice that only the hydrostatic approximation has been made to derive (9), according to the demonstration given in Appendix B to obtain the squared BVF formulation (1), where the specific moist entropy is defined by the more general and exact formula (5), with θs given by almost all the terms in (7) except that ql = qi = 0 and qt = qv.

5. The squared BVF for saturated moist air

The computations of the two partial derivatives of the density involved in the saturated formulation (1) for equation image are described in Appendix D. They are given by (D19) and (D23) and the liquid-water saturated counterpart of (9) can be written as

equation image(14)

The terms Λsw and Γsw are equal to

equation image(15)
equation image(16)

where

equation image(17)
equation image(18)

The liquid-water saturated adiabatic lapse rate is given by (16), with the derivative of T with respect to pressure computed at constant specific moist entropy and water content. The resulting value Γsw has been obtained after long computations involving the specific moist entropy defined by (5) and (7), in a way analogous to the one described in some detail in Appendix D. The specific heat cp is the moist version of it, depending on qd, qv and ql, as expressed by (D6) or in Appendix A.

The term in (14) involving the gradient of qd can be written in any of the alternative ways

equation image(19)
equation image(20)

The last formulation (20) is used in DK82 and E94. The one involving qd is the original one derived in Eq. (43) of LE74, where the superscript ‘1’ represents the dry air density. The corresponding term was written in the following way, due to the property qd = ρd:

equation image(21)

As for the unsaturated case, only the hydrostatic approximation has been made to derive (14). In particular, the specific moist entropy is defined by the more general and exact formula (5) and θs by (7), with most of the terms varying with s, qt or p.

The ice-water saturated counterparts of (14)–(18) are obtained by replacing Lvap by Lsub, rsw by rsi and qsw by qsi, including in the moist definition of cp.

In comparison with the non-saturated formulation (9) and the associated weighting factor (10), the saturated formulation (14) for N2 is modified so that the term Λv must be replaced by its saturated equivalent Λsw given by (15), with cp Rv/R replaced by Lvap/T and with the term (D1w/D2w) appearing in the adiabatic lapse rate (16) equal to 1 in the unsaturated adiabatic lapse-rate formulation (11).

6. Full comparisons with DK82 and E94

In order better to compare the present results with those published in DK82 and E94, the unsaturated and saturated squared BVF formulations given by (9) and (14) can be rewritten in terms of the gradient of temperature, by computing the vertical derivative of all the terms entering the formulation for ln(θs), with θs given by (7). It appears that most of the terms cancel out, leading to

equation image(22)
equation image(23)

The saturated squared BVF given by Eq. (13) in DK82 can be rewritten, with the notation of Appendix A, as

equation image(24)

with the same formulation for D1w as in (17), due to the equality

equation image(25)

It is worthwhile noting that all the mixing ratios were denoted by the letter ‘q’ in DK82, and that the use of the more standard letter ‘r’ is made in the present article.

The difference between (23) and (24) concerns the liquid-water saturated adiabatic lapse rate (16), defined by Eq. (19) in DK82, leading to

equation image(26)

The lapse rate computed in DK82 contains an additional term (1 + rt) and g/cp is replaced by g/cpd. Moreover, the term in the denominator of (26) may be written as

equation image(27)

and it is different from D2w given by (18) in that cp is replaced by cpd and qsw by rsw, with an additional term forming the second line of (27). All these differences make the formulations (23) and (24) more unlike than might appear at first sight.

The absence of the second line of (27) in the θs formulation for D2w given by (18) and the straight multiplication of the unsaturated adiabatic gradient by D1w/D2w in (16) indicate that the use of the θs formulation eventually leads to a more compact and more logical definition of the saturated adiabatic lapse rates.

The liquid-water saturated adiabatic lapse rate is defined in E94 as

equation image(28)

The E94 formulation contains the same additional term (1 + rt) as in the DK82 formulation (26), but with cp replaced by equation image instead of cpd. The term in the denominator may be written as

equation image(29)

In comparison with the θs formulation D2w given by (18), the E94 formulation DEM contains an additional third term, with qsw replaced by rsw and cp by equation image.

The equivalent of (23) or (24) is not mentioned in E94. However Emanuel, like Durran and Klemp, tried to express the moist and saturated value of the squared BVF in terms of conservative variables, with the same quantity qt representing the conservation of the dry air or the total water species, but with definitions of the moist entropy that are different from the one retained in the present article, depending on θs given by (7).

The moist entropy-like function appearing in DK82 was expressed in terms of the quantity cpd ln(θq), with θq defined by

equation image(30)
equation image(31)

It is worthwhile noting that, from (30) and provided the correction term depending on (T/T0) is a small one (valid in the lower troposphere where TT0 and above where rt tends to 0), θq is almost equivalent to the equivalent potential temperature θE.

The corresponding saturated value of the squared BVF is given by Eq. (21) in DK82. It may be written as

equation image(32)

The lapse rate ΓDK is given by (26) and (27).

The moist entropy-like function appearing in E94 will be denoted by s in the present article. It is different from the θs specific entropy formulations (C2) and (D4) in that Emanuel, like P11, has considered an entropy ‘per unit mass of dry air’ and not ‘per unit mass of moist air’. As a consequence, the reference values are not derived from the Third Law in E94 and P11, and the corresponding term Λr does not appear as such.

More precisely, the non-saturated and saturated versions of the moist entropy s = s/qd = s/(1 − qt) = (1 + rt) s are defined in E94 by

equation image(33)

The liquid-water saturated version of (33) is obtained by replacing rv by rsw and with a relative humidity of 100% leading to e/esw = 1, and therefore to a cancellation of the last term. It is possible to compare the saturated version equation image with (5)–(7) by using the properties pd = p/(1 + η rv) and rsw = rtrl, leading to

equation image(34)
equation image(35)

The general features of θs and θ are similar. The main difference concerns the first lines of (7) and (35), where the term Λr ≈ 5.87 in θs is replaced by Lvap/(cpd T) ≈ 9 in E94. The other differences concern the specific contents, which are replaced by the mixing ratios, and the second line of (35), where several terms in the second and third lines of (7) do not appear.

The saturated squared BVF corresponding to equation image is derived in E94. It is equal to

equation image(36)

with ΓEM given by (28) and (29).

The comparisons between (32) or (36) and the present formulation (14) show the following:

  • the last terms −(1+rt)−1∂rt/∂z in the DK82 and E94 formulations and the term ln(qd)/∂z in (14) are the same, due to the properties (19)–(20);

  • the moist lapse rates ΓDK and ΓEM are different from the one (16) obtained with θs, as explained above;

  • both ΓDK and ΓEM are divided by (1 + rt) in (32) and (36), removing the impact on equation image and equation image of the same extra term (1+rt) included in these adiabatic lapse rates;

  • the moist entropy functions are not the same, with s and cpd ln(θs) different from s or cpd ln(θq), implying vertical gradients different from those in the E94 and DK82 formulations (32) and (36);

  • the second line of (14), which represents a new term consistent with the new formulation for the specific moist entropy and θs, does not appear in the DK82 formula (32) and is only partially present in the E94 formula (36).

The formulation (36) has been expressed in E94 with the hope of managing vertical gradients of conservative variables only. It has been concluded that ‘cloudy air is stable if moist entropy increases upward and total water decreases upward’. This is true only if the moist entropy is accurately represented by s in E94. The same property holds for the DK82 formulation (32), as far as the moist entropy is accurately represented by cpd ln(θq).

A similar stability analysis may be applied to the present formulation (14), with the same stable feature valid for equation image on the first line of (14) if θs and qd increase upward, but with the presence of a non-negligible contribution in the second line, of the opposite sign to ln(qd)/∂z > 0 and larger in absolute value for classical atmospheric conditions. The novel aspects of this contribution, without any equivalent in the DK82 and E94 formulations (32) and (36), will be studied in detail in the next section.

The second line of (14) almost disappears in E94 and does not exist in DK82. It is possible to explain this feature by comparing the potential temperature θs given by (7) with the E94 formulation θ given by (35). Clearly, the term Λr in θs is replaced by Lvap/(cpd T) ≈ 9 in θ. This modification would transform the second line in (14) into a term depending on rsw Lvap /Tcpd Λsw. It is a residual quantity that is much smaller than Lvap /Tcpd Λr and this explains why the impact of the new term Λr depending on the absolute values of the reference partial entropies is important in the second line of (14) for equation image.

7. Impact of gradients of qt and of moist entropy formulations

The remarks and comparisons mentioned in the previous section indicate that the DK82 and E94 formulations possess two potential conceptual drawbacks with respect to the present proposal.

First, the DK82 and E94 moist lapse rates (26) and (28) should not contain the term (1 + rt). This term is a consequence of the moist entropy being defined ‘per unit mass of dry air’ in DK82 and E94, corresponding to the transformation of any specific value ψ into ψ = (1 + rt) ψ = ψ/(1 − qt) = ψ/qd. It is important to notice that, even if the moist lapse rates are multiplied by (1 + rt), this has no impact on equation image and equation image, since the lapse rates are divided by the same factor in (32) and (36). The adiabatic lapse rates of a parcel in a moist atmosphere should however have a unique definition, and the specific moist entropy based on θs and (7) appears to be a more general way to obtain it, leading to the simple shape of Γsw given by (16)–(18).

Second, it seems that one of the constraints for determining or choosing the moist entropy formulations s/qd or cpd ln(θq) may be to express the squared BVF as a sum of two terms only, with a first term depending only on the gradient of moist entropy formulations plus another term depending only on the gradient of qt, but being exactly equal to the prescribed value −g/(1 + rt)(∂rt/∂z) given by (20). It is indeed the result (32) obtained in DK82 and almost achieved in E94, where a partial second line still exists in (36).

The specific moist entropy has a unique formulation, since it is a thermodynamic state function. In this respect, different moist entropy formulations must lead to different sets of curves in the conservative variable diagram plotted in Figure 1. The solid lines and the dashed lines coincide for the dry air limit at the bottom of the diagram, due to the global shift by the amount of the dry air standard value equation image for s.

Figure 1.

The same conservative variable diagram as in PS10 with total specific water content plotted against moist entropies. The temperature T(s,qt,p1) is plotted for p1 = 900 hPa. The diagram is separated by the increasing saturation curves located in the centre of it, with non-saturated and saturated regions located below and above the curves, respectively. The dashed lines represent equal values of T(s,qt,p1) computed with the PS10 moist entropy s(θ) given by (33)–(35), depending on the non-saturated or saturated conditions, respectively. The solid lines correspond to equal values of T(s,qt,p1) computed with the specific moist entropy formulations s(θs) given by (C2) or (D4), depending on the non-saturated or saturated conditions, respectively. The θs entropy formulation is plotted as equation image on the x-axis, with equation image J K−1 kg−1. The lines of equal temperature are labelled every 10 K. The black circles represent the changes in s or s due to the moistening of an initially dry-air parcel of atmosphere, with the same increase of qt of 5 g kg−1 and undergoing isothermal processes following lines of equal value of either T(s,qt,p1) or T(s,qt,p1). The black stars represent the changes in moist entropies due to the moistening of an initially saturated parcel by the same amount of 5 g kg−1 and undergoing the same kind of isothermal processes.

The differences between lines of equal value of T become more and more important as qt increases. The saturation curves are also different, depending of the use of s(θs) versus s(θ) formulations. The dashed lines are of almost equal values of s in the saturated region, whereas the specific moist entropy s decreases as qt increases in the saturated region. These differences demonstrate that the way in which the moist entropy is defined may generate important differences in the physical interpretation.

In fact, it is unlikely that some arbitrariness may exist in the possibility to change the formulation of the specific moist entropy. It is the difference in the formulation of the moist entropy that generates the large impacts on the definition of the moist squared BVF and the different ways to write the terms depending on the gradient of qt. Moreover, even if the squared BVF values eventually remain close to each other, different values for the changes in moist entropy correspond to different physical meanings.

In particular, for isothermal and isobaric transformations where qt increases by the same given amount of 5 g kg−1, the black circles and stars represent the changes in moist entropies associated with isothermal and isobaric transformations. The changes in moist entropies are clearly different in PA10's version and the present one. In saturated conditions, the change of entropy almost cancels out with s(θ), whereas the specific moist entropy exhibits a large decrease with s(θs). It may be considered that the almost isentropic feature appearing for the isotherms above the saturation curve with the formulations of PA10 and E94 is a very special case that is not explained by theory, nor supported by observations.

The reference values are determined in s(θs) from the Third Law and Λr ≈ 6 in (7) is replaced by Lvap/(cpd T) ≈ 9 in (35). It is worthwhile noting that when Λr is arbitrarily set to 9 in the θs definition (7), the solid lines are almost superimposed on the dashed lines in Figure 1 (result not shown). This indicates that the Third Law used to compute the special value of Λr is a key part of the definition of s(θs) and that the logic of PA10 and E94 is of a different kind.

It is therefore important to justify the present formulation, with the above prescribed term separated from the other terms in the second line of (9) and (14) even if those also depend on the gradient of qt. In fact, this prescribed term was introduced first in LE74 as a correction to older formulations, in the form g∂ln(qd)/∂z given by (21). This correction term also appears in the DK82 moist version in the equivalent form (20), where it was still considered as an additional term.

By analogy with the ideas published in Pauluis and Held (2002), this correction term may be interpreted as a modification to the vertical stability, represented here by equation image, and corresponding to a conversion between the kinetic and potential energy due to the work required to ensure the vertical transport of water species.

Hence, the correction term given by (20) logically appears in the DK82 and E94 conservative variable formulations (32) and (36) and also in the first lines of the present formulations (9) and (14) for equation image and equation image, respectively, but given by the equivalent form (21). The correction term must not be included in the specific moist entropy term, nor be regrouped with the second line of (9) and (14).

The computation of the moist squared BVFs in terms of the vertical gradient of θv is derived in Appendix E, showing that the moist squared BVF based on the specific moist entropy s(θs) is not exactly proportional to ∂θv/∂z for a moist but non-saturated atmosphere.

An interesting feature suggested by the second lines of (9) and (14), in which the moist squared BVF is based on the θs approach, is that a continuous transition may exist between the two unsaturated and saturated regimes. An example of this kind of transition is described in Appendix F.

Summing up, the important new result derived in the present article is that the specific moist entropy (and θs) really verify the conservative property associated with the second principle. The consequence is that second lines must exist in both the non-saturated version (9) and the saturated one (14). The physical meaning for these extra terms can be found in the results obtained in M11, where the vertical profile for the specific moist entropy s is observed to be almost a constant for marine stratocumulus, in spite of existing vertical trends for the B73 variables qt and θl. These results are true for both clear-air (unsaturated) and in-cloud (saturated) moist regions.

This means that, at least for marine stratocumulus, the sign of equation image and equation image is not controlled by the vertical gradient of specific moist entropy but almost entirely by the vertical gradient of qt. More precisely, the vertical profiles of qt impact on equation image and equation image not only via the ‘water lifting’ contributions located in the first lines of (9) and (14), but also via the terms located in the second lines, where ‘expansion work’ and ‘latent heat’ effects are accompanied by the new impact corresponding to the pure entropy terms Λr, Λv and Λsw.

The second lines of (9) and (14) are of opposite sign. It is possible to explain this result by comparing

equation image(37)

The result is that Λr is almost in a two-thirds/one-third position between Rv/R and Lvap/(cpdT) in terms of the control parameter C (see Appendix F), leading to positive values for Lvap/Tcpd Λr in the second line of equation image and to negative values for the corresponding term in the second line of equation image. This result may be put into context with the property illustrated in Figure 2, where θs is almost in a two-thirds/one-third position between θl and θE (see also M11).

Figure 2.

Comparison of θs (in a central position) with other potential temperature formulations for the grid-cell average of the RF03B profiles of the FIRE-I datasets. The two profiles on the left are θl and θv given by (40) and (3). The four last profiles on the right correspond to four formulations of θE (B73, E94, Bolton (1980) and a formulation used in ARPEGE). The vertical profile of the approximate version (39) is located close to the θs curve and is labelled by (θs)1.

8. Approximate versions for the moist squared BVF

It is shown in M11 that the specific moist entropy defined by (5) can be accurately approximated by

equation image(38)

The reference entropy is still given by (6), but the specific moist entropy potential temperature θs is approximated by (θs)1 given by the first line on the right-hand side of (7), leading to a liquid water version, which may be written as

equation image(39)

where

equation image(40)

is the B73 liquid-water potential temperature.

The specific moist entropy s[(θs)] and its approximate version s1[(θs)1] are compared in Figure 2 with other usual moist potential temperatures. Clearly, the replacement of θs by (θs)1 is a good approximation, with errors (θs)1θs much smaller than the observed large differences between θs and θl, θv or θE. Moreover, the errors are almost constant along the vertical and they should not largely impact on the computations of the moist squared BVF.

The computations of the moist squared BVF and the moist adiabatic lapse rates can be realized through the replacement of s by s1. The corresponding approximate non-saturated and saturated versions of the squared BVF may be written as

equation image(41)

and

equation image(42)

The term D1w is still given by (17) but D2w is replaced by

equation image(43)

A new term depending on ql appears in D2wl, with the specific heat cp in (18) replaced by cpd in (43) and equation image given by (D7).

The last term involving equation image and ql is somehow similar to the last terms in the DK82 and E94 formulations (27) and (29). These terms are of the order of 1.8 rsw in DDK, 4.2 rl in DEM and 11 ql in D2wl. It means that when the moist entropy is based on formulations different from s(θs), the compact feature obtained in (18) for D2w is modified. It is indeed slightly different from D2wl and the approximation s1[(θs)1] of this section, or from DEM and E94's formulation s(θ). The modifications of DDK are more important for DK82's version cpd ln(θq), since typical values for rsw are much larger than those for rlql.

The non-saturated and saturated lapse rates may be written as

equation image(44)

and

equation image(45)

where

equation image(46)

The difference between D1w and D3w is that the virtual temperature Tv in (17) is replaced by the actual one T in (46).

Comparison of the conservative variable diagrams plotted in Figures 1 and 3 shows that the impacts of the difference between the specific moist entropy formulation s[(θs)] and the approximate version s1[(θs)1] are much smaller than the impacts of the use of the moist entropy formulation of Pauluis and Schumacher (2010, hereafter PS10) s(θ). This allows the possible use of s1[(θs)1] as an accurate approximation of s[(θs)], whatever the non-saturated or (over-)saturated conditions may be.

Figure 3.

The same conservative variable diagram as in Figure 1, but with the exact moist entropy formulation s(θs) compared with the approximate one s1[(θs)1] and with extended ranges for the variations of the moist entropies and of qt. The solid lines correspond to constant values of T(s,qt,p1) computed with s(θs) given by (C2) and (D4). The dashed lines correspond to constant values of T(s1,qt,p1) computed with the approximate version s1[(θs)1] coming from (38) and (39). The entropies are plotted as equation image and equation image on the x-axis, with equation image J K−1 kg−1. Lines of equal temperature are labelled every 10 K.

As an example of a possible application, it is possible readily to convert the ‘bridging relationships’ (F1)–(F4) to the case of θs ≈ (θs)1. If cpd replaces cp in the definition of F(C) and if the last term of (43) appears in the lower case expression of M(C), the equivalent of (F4) is then simply expressed by

equation image(47)

This more compact formulation allows us better to understand the purpose and limitations of the introduction of C as control (or transition) parameter for the bridging step synthetically described by (47).

Potential applications of a formula like (47) may correspond (pending other suggestions left to interested readers) to

  • the computation of a N2-linked physical quantity like the conversion term of turbulent kinetic energy into other forms of energy;

  • the calculation of a Richardson number (or any 2D or 3D related quantity);

  • some more complex computations, like those of the reduced complexity model for interactions between moist convection and gravity waves described in Ruprecht et al. (2010) and Ruprecht and Klein (2011). There, C would represent de facto the area fraction of cloudy air in horizontal slices.

In all cases, the question of the definition (or the parametrization) of the control parameter C becomes a central one and this issue is likely to take a differing shape from case to case. If seeking full complexity, C should not be confused with the proportion of saturated air within the considered air parcel. There are two reasons for this. First, as explained in Appendix F, there is no reason to consider N2(C) or equation image as a C-weighted linear interpolation between the extreme cases of fully unsaturated and fully saturated conditions. Second, in most conditions there would exist (partly) organized motions differentiating between the mean dynamical behaviour of the clear-air and cloudy patches of the considered air parcel respectively. Nonetheless, we may postulate a monotonic dependence of C on the above-mentioned proportion.

Despite the weakness linked to the generally heuristic character of the definition of C, two additional remarks support the potential use of (F4) or (47).

First, in the already mentioned case of FIRE-I marine Stratocumulus clouds, there is hardly any gradient of specific moist entropy between clear-air and cloudy patches in Figure 4(a), and most of the so-called subgrid transport of qt is ensured by turbulent motions and not by partly organized compensating motions between these patches. Hence, viewing here C as a kind of subgrid cloud cover becomes rather legitimate, if one indeed attributes the nonlinear part of the N2(C) or equation image behaviour to the existence of the above-mentioned (small) partly organized transport.

Figure 4.

A study of the FIRE-I (RF03B) stratocumulus (see M11). The vertical profiles of θl, θv, (θs)1, θs, θE and θq are depicted in (a), from left to right. The vertical profiles of qt and 10 × ql are depicted in (b). In both (a) and (b) the solid (dashed) lines represent in-cloud (clear-air) and saturated (unsaturated) conditions. Only in-cloud values of θq are plotted, since the formulation (30) is not valid for clear-air (unsaturated) conditions. The vertical profile of the clear-air approximate formulation (θs)1 is located close to the clear-air profile for θs, with an almost constant bias of about −0.4 to −0.6 K (see M11). Units are in K for the potential temperatures and in g kg−1 for the water contents. The budgets of the moist squared BVF equations are depicted in (c)–(f). The total budgets (heavy solid lines) are split into terms depending either on the lapse-rate or gradients of potential temperatures (solid lines) or on the gradients of the water contents qv or qt (dashed lines). The usual lapse-rate budget (22) of the clear-air (unsaturated) squared BVF formulation is depicted in panel (c). The standard formulation (E1) for equation image expressed in terms of the gradient of θv is depicted by the long-dashed line. The new clear-air budget (9) using the (unsaturated) θs specific moist entropy formulation is depicted in panel (d). The DK82 squared BVF formulation (32) valid for in-cloud (saturated) conditions and using θq is depicted in panel (e). The new in-cloud (saturated) θs specific moist entropy formulation (14) is depicted in panel (f). All the squared BVF values are multiplied by a factor of 104, with units in s−2.

Second, the fact that one single parameter is sufficient to obtain a monotonic, general and consistent transition between unsaturated and saturated situations is a welcome step for applications seeking a robust and simple behaviour.

In summary, the proposed transition formulae need to be used with a lot of care (especially for the estimation of C): there is at least one case in which they should be directly appropriate, while in other cases they may be useful because of their simplicity (one control parameter alone) and their monotonic character, for lack of other alternatives in the approach to practical problems.

9. Some numerical applications

A numerical application is presented in this section by using the same RF03B FIRE-I observations as in M11, except with the additional constraint that qvqsw if ql > 0.1 g kg−1 and ql ≡ 0 if ql < 0.1 g kg−1 . The profiles have been slightly filtered vertically, in order to give smoother profiles and less noisy vertical gradients. The same average profiles are used for all the formulations of N2.

The vertical profiles of the basic variables are depicted in Figures 4(a) and (b), where the θq curve given by (30) appears to be similar to the saturated θE one, with θs ≈ (θs)1 ≈ 304.5 K in a two-thirds position between θlθv ≈ 288 K and θEθq ≈ 312 K, as already indicated in figure 11(b) of M11.

The clear-air profiles of Betts variables θl and qt are clearly different from the associated in-cloud profiles in the upper planetary boundary layer (PBL) entrainment region. The same feature is valid for the θv and θq curves (the two profiles used in DK82). In contrast, the clear-air and in-cloud vertical profiles of θs are almost superimposed, illustrating the full mixing in specific moist entropy within the stratocumulus, as already mentioned in M11.

Moreover, the θq and θE profiles are almost similar but far from the θv curve, with almost opposite vertical gradients. This must correspond to a less continuous feature between the clear-air and in-cloud formulations of N2 with the standard DK82 approach, where the non-saturated budget of N2 is based on θv and the saturated one is expressed in terms of θqθE.

Let us comment on the Figures 4 (c)–(f). For the standard clear-air formulation (22), which uses the lapse-rate approach, the budget of the squared BVF is dominated in (c) by the thermal component, with a much smaller water content component. For the new formulation in (d), which uses the conservative variable (θs,qt) approach, the (total) value of N2 is almost the same as for DK82, but the clear-air budget is made of large and compensating thermal versus water content components (within the whole moist PBL and the dry air above as well).

A more detailed analysis shows that some numerical differences exist between 650 and 800 m (thin lines have been added at 700 and 800 m), where the total N2 budgets (thick solid lines) are not the same in (c) and in (d). It appears that the vertical profile of T (not shown) exhibits more noisy and uneven vertical shape than the vertical profiles of qv, leading to gradients of θs that are easier to determine than the lapse rates. Since the standard formulation (E1) for equation image (long-dashed line in (c)) is very close to the (θs,qt) approach (thick line in (d)), it may be concluded that the differences are due to less accurate evaluations of the stability feature using the lapse rate method than with the other methods based on virtual or specific moist entropy potential temperatures.

The in-cloud budgets presented in (e) and (f) show that the total values of saturated N2 are almost the same for the DK82 formulation with ln(θq) as for the new formulations with ln(θs). However, for the new formulation in (f), the water content component is of opposite sign and is more ‘neutral’ than in (e), i.e. it is closer to 0 at each level. These differences are the consequences of the second line in (14), which does not exist in (32), leading to a different partitioning of the budgets of N2.

10. Conclusions

Both non-saturated and saturated versions of the moist squared BVF (N2) have been computed in terms of the vertical gradients of the moist natural conservative variables, namely the specific content of dry air (or total water) and the specific moist entropy. The latter has been defined in terms of the specific entropy potential temperature (7) for θs introduced in M11, differently from the moist entropies and potential temperatures already defined in DK82, E94, P08 or P11.

Comparisons with the previous results published in DK82 and E94 show that the adiabatic lapse rates are different. The conservative variable diagrams published in P11 are also modified if the present θs formulation for the specific moist entropy is used, with possible different physical properties. A new small counter-gradient term appears when the new non-saturated version of N2 is written in terms of the vertical gradient of the virtual potential temperature.

Numerical applications made with the FIRE-I data sets indicate that there is little difference for the (total) values of N2. Larger impacts are observed if the budget of N2 is partitioned into a sum of separated terms depending on gradients of s and qd (first lines) and qt (second lines), with weighting factors significantly different from the ones obtained with DK82 or E94 moist entropies.

It is possible to replace θs by the approximate version (θs)1 and to derive a corresponding approximate formulation for N2. A continuous transition is suggested between the new non-saturated and saturated versions of N2, leading to the possible definition of a control parameter C valid for both (θs)1 and θs formulations.

It is a kind of paradox that the complexity of the specific moist entropy defined with the full formulation of θs should lead to rather simple and compact formulations for equation image and equation image. In particular, the terms DDK and DEM involved in the DK82 and E94 computations of the moist, saturated, adiabatic lapse rate and the associated squared BVFs are more complicated than with the formulations Γsw and equation image based on θs. Additional small terms depending on the liquid-water content appear in DDK and DEM, and they disappear in Dsw.

An explanation for this paradox could be found in the complex moist basic formulae like (D6) and (D7), which define cp or Lvap(T), among others. They both depend on the thermodynamic properties of water species in such a way that the full specific entropy formulation (7) for θs seems to be required in order to arrive at a cancellation of all small terms. If approximations are made in the definitions of the moist entropies, like the hypothesis of zero dry-air and liquid-water reference entropies in E94, or in the definitions of R, cp or Lvap, the cancellation of the small terms is incomplete. This is true, for instance, for the θq and θ used as starting points to compute N2.

It may be worthwhile to note that the approximation of θs by (θs)1 generates different but still simple and compact versions of equation image and equation image. As for the DK82 and E94 versions, the corresponding term D2wl contains an additional small term depending on ql, but the same continuous transition with the same parameter C is obtained between equation image and equation image as between equation image and equation image. In this respect, it may indicate that the approximation of θs by (θs)1 is of smaller impact than the use of θq or θ.

Acknowledgements

The authors are most grateful to D. Mironov for initially insisting on the potential importance of this work as well as for further exchanges, to R. De Troch for stimulating discussions and to O. Pauluis for his encouragement to go forward with the complex analytical task. Both Rupert Klein and the other anonymous reviewer have clearly helped to improve the scope and content of the article by requiring a more general and more ambitious approach, something eventually much appreciated by the authors. Most of this work was completed in the spirit and framework of the EU-funded COST ES0905 action.

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

Appendix A. List of symbols and acronyms

(∂/∂z)parGradients computed for the parcel
(∂/∂z)envGradients computed for the environment
BVFBrunt–Väisälä Frequency
C, C0control parameters in Appendix F
cpdspecific heat for dry air    (1004.7 J K−1 kg−1)
cpvspec. heat for water vapour    (1846.1 J K−1 kg−1)
clspec. heat for liquid water    (4218 J K−1 kg−1)
cispec. heat for ice    (2106 J K−1 kg−1)
cpspecific heat at constant pressure for moist air,
 = qdcpd + qvcpv + qlcl
 = qd(cpd + rvcpv + rlcl)
equation image= cpd + rswcpv (E94's formulation)
D1wa shortcut notation, like D2w, DDK, DEM, ...
δ= Rv/Rd − 1 ≈ 0.608
η= 1 + δ = Rv/Rd ≈ 1.608
ε= 1/η = Rd/Rv ≈ 0.622
κ= Rd/cpd ≈ 0.2857
γ= ηκ = Rv/cpd ≈ 0.46
λ= cpv/cpd − 1 ≈ 0.8375
ethe water-vapour partial pressure
esw(T)partial saturating pressure over liquid water
erthe water-vapour reference partial pressure,
 with er = ews(T0) ≈ 6.11 hPa
FIREthe First ISCCP Regional Experiment
F(C)a function of C, like M(C)
ggravity's constant (9.80665 m s−2)
Γnsthe lapse rate (−∂T/∂z / unsaturated)
Γswthe liquid-water saturated version of Γns
ISCCPInternational Satellite Cloud Climatology Project
Λr= [(sv)r − (sd)r]/cpd ≈ 5.87
Λvan additional term to Λrsw as well)
Lvap(T)= hvhl: latent heat of vaporization
Lvap(T0)= 2.501 × 106 J kg−1
Lsub(T)= hvhi: latent heat of sublimation
Lsub(T0)= 2.835 × 106 J kg−1
equation imagea latent heat shortcut notation
N2squared BVF notations (equation image, equation image, equation image, ...)
PBLplanetary boundary layer
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
ψa dummy variable (section 7 and Appendix B)
qd= ρd/ρ: specific content for dry air
qv= ρv/ρ: specific content for water vapour
ql= ρl/ρ: specific content for liquid water
qi= ρli/ρ: specific content for solid water
qswspecific content for saturating water vapour
qt= qv + ql + qi: total specific content of water
rv= qv/qd: mixing ratio for water vapour
rl= ql/qd: mixing ratio for liquid water
ri= qi/qd: mixing ratio for solid water
rrreference mixing ratio for water species, with
 ηrrer/(pd)r, leading to rr ≈ 3.82 g kg−1
rswmixing ratio for saturating water vapour
rt= qt/qd: mixing ratio for total water
ρdspecific mass for dry air
ρvspecific mass for water vapour
ρlspecific mass for liquid water
ρispecific mass for solid water
ρspecific mass for moist air
 = ρd + ρv + ρl + ρi
Rvwater vapour gas constant --&equals; (461.52 J K−1 kg−1) Rd dry-air gas constant (287.06 J K−1 kg−1)
Rvwater-vapour gas constant (461.53 J K−1 kg−1)
R= qdRd + qvRv: gas constant for moist air
 equation image
sthe specific moist entropy associated with θs
s1the specific moist entropy associated with (θs)1
srefa reference specific entropy
sdspecific entropy for the dry air
svspecific entropy for the water vapour
slspecific entropy for the liquid water
s*a moist entropy (E94)
(sd)rreference values for the entropy of dry air,
 at Tr and (pd)r
(sv)rreference values for the entropy of water vapour,
 at Tr and er
equation imagestandard specific entropy for the dry air
 (value at T0 and p0: 6775 J K−1 kg−1)
equation imagestandard specific entropy for the water vapour
 (value at T0 and p0: 10320 J K−1 kg−1)
Tlocal temperature
Tvvirtual temperature associated to θv
TρE94's version for Tv
Trthe reference temperature (TrT0)
T0zero Celsius temperature ( = 273.15 K)
θequation image: potential temperature
θ*moist entropy potential temperature (E94)
θqmoist entropy potential temperature (DK82)
θEequivalent potential temperature
θvvirtual potential temperature
θlliquid water potential temperature
θsspecific moist entropy potential temperature (M11)
s)1approximate version of θs
zvertical coordinate

Appendix B. General squared BVF formulations

The method for computing the moist value of N2 is usually based on the classical approach of DK82, where the adiabatic changes of the density of the parcel are evaluated and compared with the corresponding values of the environment, leading to

equation image(B1)

Let us consider the method based on the material published in PA08 and PCK10, where it is stated that any thermodynamic variable ψ can be expressed in terms of the entropy s, the total water content qt and the pressure p alone. This is true in particular for ψ representing any of the temperature (T), the specific volume (α), the density (ρ), the water contents (qv, ql, qi) or the buoyancy (B), leading to ψ(s,qt,p).

It is indeed possible to use the set of three independent variables (s,qt,p) if it is assumed that a parcel of moist atmosphere is either saturated (with qv equal to its saturated value and with existing condensed water equal to qtqv) or non-saturated (with no condensed water and qt = qv). In this way, the condensed water contents ql and qi no longer appear as independent variables of the system and they must be derived from the information given by (s,qt,p), with either liquid water for T > T0 or solid water for T < T0.

The property (1) can be derived through a short mathematical method, starting with (B1) rewritten as

equation image(B2)

From the chain rule, the gradient of the density ρ(s,qt,p) is equal to

equation image(B3)

If hydrostatic conditions prevail, then dp = −ρgdz is applied twice in the last terms of (B3), this last term being equal to

equation image(B4)

The property (1) is obtained with (B4) inserted into (B2).

Appendix C. The unsaturated moist squared BVF

The properties ql = qi = 0 and qt = qv are used to derive the unsaturated moist air version of the state equation (3) and the virtual temperature definition (3), resulting in

equation image(C1)

The properties ql = qi = 0, qt = qv and rt = rv are used to transform the definitions (5) and (7) to express the entropy for unsaturated moist air as

equation image(C2)

The first partial derivative of ρ with respect to s (at constant values for p and qt) is computed from (C1) and with T = T(s,qt,p), leading to

equation image(C3)

The partial derivative of T with respect to s (at constant values for p and qt) is obtained by computing the derivative of (C2) with respect to s and with T = T(s,qt,p) (i.e. involving only the two terms in the first line on the R.H.S.), leading to

equation image(55)
equation image(C5)

From (C3) and (C5), the first partial derivative involved in the formulation (1) for the non-saturated version of equation image is equal to

equation image(C6)

The second partial derivative of ρ with respect to qt (at constant values for p and s) is computed from (C1) and with T = T(s,qt,p), leading to

equation image(C7)

The partial derivative of T with respect to qt (at constant values for p and s) is obtained by computing the derivative of (C2) with respect to qt and with T = T(s,qt,p), to arrive at

equation image(C8)

where Λv is given by (10). This term is equal to 0 for the reference conditions T = Tr, p = pr and rv = rr and, as such, it is expected to be a corrective term to Λr.

It is important to notice that the third term −γln(rv/rr) on the right-hand side of (10) becomes infinite when rv tends to 0, leading to an ill-defined dry-air version of Λv. However, the contribution to the moist squared BVF is proportional to the product of Λv by the gradient ∂qv/∂z. Accordingly, the dry-air limit must be computed within a given dry region around q0 = qv(z0) = 0 and where qv(z) − q0 is positive and very close to zero, leading to a first-order formulation of qv(z) proportional to (zz0)2 and to a gradient proportional to zz0, and thus to equation image. This result shows that the product Λv (∂qv/∂z) contains a term varying as equation image which has 0 as a limit when qv and rv = qv/(1 − qv) tend to zero.

The computations needed for deriving (C8) and (10) are rather long. They have been obtained with the help of the properties qt = qv, rt = rv, rt = qt/(1 − qt), ∂rt/∂qt = (rt/qt)2, rv = qv(1 + rv), κ η = γ, δ = η − 1 and in particular with the identity

equation image(C9)

From (C7) and (C8), the second partial derivative involved in the formulation (1) for the non-saturated version of equation image is equal to

equation image(C10)

In order to be consistent with the saturated version derived in the next Appendix, (C10) can be written differently. The last term (T/Tv) is replaced by (1 + η rv) (T/Tv) and the additional part η rv (T/Tv) is then subtracted from the bracketed term of (C10), together with the following identities:

equation image(C11)
equation image(C12)

The result is

equation image(C13)

Appendix D. The saturated moist squared BVF

The saturated squared BVF is computed in this Appendix only using the liquid water content, since the hypotheses retained in Appendix B do not allow the possibility of having liquid and solid species in a parcel of moist air at the same time. In fact, the same hypothesis is made for the derivation of the specific moist entropy formulation (5), with the sum Lvap ql + Lsub qi to be understood as Lvap ql or Lsub qi, depending on T > 0 or T < 0, with either ql0 or qi0, respectively. The ice content formula can be derived through symmetry properties: Lvap replaced by Lsub, ql by qi and rl by ri.

The properties qi = 0, qv = qsw(T,p,qt) and ql = qtqsw(T,p,qt) are used to derive the following saturated moist air version of the state equation (2), to arrive at

equation image(D1)
equation image(D2)
equation image(D3)

The liquid water saturated entropy can be written as

equation image(D4)

where Lvap only depends on T and where, from (D2), rsw = qsw/(1 − qt) only depends on T and p.

The computations of the partial derivatives of ρ with respect to s or qt are more complicated than for the unsaturated cases and the following properties must be taken into account:

equation image(D5)
equation image(D6)
equation image(D7)
equation image(D8)
equation image(D9)
equation image(D10)
equation image(D11)
equation image(D12)

The derivatives at constant pressure of the saturating specific content and mixing ratio are equal to

equation image(D13)
equation image(D14)

Moreover, chain rules are applied to the derivatives of qsw(T,qt,p) and rsw(T,p), leading to

equation image(D15)
equation image(D16)

The previous properties allow the computation of the first partial derivative of (D1) with respect to s (at constant values for p and qt), leading to

equation image(D17)

where D1w is given by (17).

The partial derivative of T with respect to s (at constant values for p and qt) is obtained by computing the derivative of (D4) with respect to s and with T = T(s,qt,p), to arrive at

equation image(D18)

where D2w is given by (18).

From (D17) and (D18), the first partial derivative involved in the liquid-water saturated formulation (1) for equation image is equal to

equation image(D19)

The difference from the non-saturated case (C6) is the extra term (D1w/D2w).

The second partial derivative of ρ with respect to qt (at constant values for p and s) is computed from (D1) with T = T(s,qt,p) and with the chain rule applied to the derivative of qsw(T,qt,p), leading to

equation image(D20)

The term D1w is again given by (17).

The partial derivative of T with respect to qt (at constant values for p and s) is obtained by computing the derivative of (D4) with respect to qt, with T = T(s,qt,p) and with chain rules applied to qsw(T,qt,p) and rsw(T,p), leading to

equation image(D21)
equation image(D22)

The term D2w is again given by (18). The term Λsw is equal to the non-saturated version of Λv given by (10), but expressed for the saturated conditions rv = rsw, leading to (15).

Even if the results are compact and consistent at first sight with (C8) and (10), the computations made to derive (D22) and (15) are rather long. They have been obtained by taking into account the properties qi = 0, qv = qsw(T,qt,p), qt = qsw + ql, together with the results (D2)–(D14).

From (D20) and (D22), the second partial derivative involved in the formulation (1) for equation image is thus equal to

equation image(D23)

Appendix E. Comparisons with the buoyancy formulations

The specific moist entropy formulations (9) and (14) have been obtained without any approximation, except the hydrostatic one used in Appendix B to demonstrate the generic BFV formula (1).

The advantage of the formulations (9)–(14) is that they are expressed in terms of the gradients of the more general conservative variables, i.e. the specific moist entropy and the chemical fractions of the parcel (or equivalently the concentrations in dry air or total water content).

It is generally accepted that the non-saturating squared BVF may be defined by

equation image(E1)

It may be important to try to express the specific moist entropy formulations (9) and (14) in terms of the vertical gradient of the buoyancy potential temperature θv given by (3).

Let us derive the non-saturated moist air formula (9) in terms of the vertical gradient of qv = qt and θv by computing both the vertical gradient of θv and the vertical gradient of the specific moist entropy (5), with θs given by (7) and with the hydrostatic assumption, leading to

equation image(E2)
equation image(E3)

When (E3) and (E2) are inserted into the non-saturated moist air formula (9), most of the terms cancel out and the final result is

equation image(E4)

The comparison of (E1) with (E4) shows that a non-saturated counter-gradient term Γc/ns appears. It is equal to

equation image(E5)

It depends on qv and, for a typical moist PBL where qv = 10 g kg−1, θT and cp ≈ 1000 J K−1 kg−1, the values λδ ≈ 0.23 leads to Γc/ns ≈ 0.023 K km−1. This implies a contribution of −0.008 × 10−4 s−2 to equation image. It is a small term when it is compared with significant values of equation image, which are typically about 100 times larger.

Appendix F. Transitions betwen unsaturated and saturated formulations

When comparing both unsaturated and saturated moist expressions for the squared BVF, one notices a good deal of similarity: the ‘water lifting term’ is the same, the multiplying gradients are identical and there are thus only three kinds of transitions to consider. The first one is the natural expression of the terms depending on rv in the unsaturated case by the equivalent in terms of rsw in the saturated case, this being applied to both the (1 + rv) multiplicator and the Λv expression. The second one is the transition between lapse rates Γns and Γsw, i.e. the multiplication by D1w/D2w. The third one is the replacement of (cp Rv)/R by Lvap/T.

The crucial point for ensuring a complete and smooth transition between the two formulations is that, together with the logical number one transition, going from D1w/D2w to 1 and from Lvap/T to (cp Rv)/R both just require replacing (Lvap R)/(cp Rv T) by 1!

A simple way to create a generalized N2 formula on the basis of this structural symmetry between both transitions is therefore to define

equation image(F1)
equation image(F2)
equation image(F3)

to obtain

equation image(F4)

In the above set of equations the subscript ‘E’ represents the environmental value of any moist air parcel, independently of whether the conditions are fully unsaturated, partly saturated or fully saturated.

Even if the exact identity between both analytical manifestations of the transition via F(C) is a welcome result, there is some physical consistency in having the same term playing the key role in the lessening of the resistance to buoyant motions when condensation occurs on the one hand and in the replacement within the buoyancy term of gaseous density effects by latent heat release impacts on the other hand.

It can be verified that the two formulae (9) and (14) are obtained from (F4) by the two limiting cases described in Table F1. Furthermore, the product of F(C) by M(C) within the last but one term clearly shows that we have here more than a linear interpolation of N2 between the two extreme cases, as indicated by the nonlinear heavy solid curve depicted in Figure F1.

Figure F1.

The curves F(C) and M(C), defined by (F1)–(F3), are computed for ‘just-saturated’ conditions (T = 10 °C, p = 900 hPa, qv = qsw, ql = 0) and for C varying from 0 (unsaturated) to 1 (saturated). The values 0 and 1 are depicted as thin horizontal and solid lines. The third curve M(C) [ (1 + rv) Rv F(C)/R − (cpd/cp) (Λr + Λv) ] corresponds to the new extra term in the second line of (F4), with a thin line added to show that the zero value is obtained for C0 ≈ 0.55.

Table F1. The values for F(C) and M(C) for the unsaturated case defined by C = 0 and (qv, rv, Λv), and for the saturated case defined by C = 1 and (qsw, rsw, Λsw). The moist formulations of cp and R slightly depend on qd = 1 − qt, qv, ql or qi.
 C = 0C = 1
F(C)1(Lvap R)/(cp Rv T)
M(C)1D1w/D2w

In fact, the special regime that cancels the second and third lines of (F4) corresponds to the property

equation image(F5)

For the typical values Λr ≈ 5.87, Λv ≈ −0.32, R/Rv ≈ 0.622 and T = 283 K, then Lvap/(cp T) ≈ 8.6 and (Λr + Λv)/(1 + rv) ≈ 5.5, leading to the reversal value C0 ≈ (5.5 × 0.622 − 1)/(8.6 × 0.622 − 1) ≈ 0.55. This value of C0 can be associated with the approximate 5.5/8.6 ≈ 2/3 position of θs between θl and θE, as seen in Figures 2 and 4(a).

Lastly, on top of the contribution to obtaining θs, (F4) clearly separates the roles of qt and qv in the rest of the N2 expression. The total content qt (or its complement to one qd) is present only through its vertical gradient, a key quantity in our way of obtaining the squared BVF. The water-vapour content qv is present in M(C) only via DC and in the second- and third-line parentheses via the moist definition of cp, (1 + rv) and Λv, under the implicit understanding that it will equal qsw for C = 1 in all these occurrences. One should nevertheless realize that, numerically speaking, the above implicit assumption may not be perfectly obeyed without this having any bad consequences for the computation of a generalized moist squared BVF according to Eq. (F4). As a consequence of this continuity-rule-obeying sorting-out of the various roles of moisture quantities, (F4) does not contain any explicit calculation linked to water phase changes (no presence of qsw or ql for instance).

There are, however, some caveats associated with the use of N2(C). First, it is clear that except in the extreme homogeneous cases corresponding to C = 0 and C = 1, the squared BVF loses its original meaning associated with the period of natural oscillations of an air parcel displaced along the vertical. Second, it is no longer possible to find an equivalent of (1) leading to (F4), since the simplifications allowing us to express T as a function of s, qt and p (as used in Appendices C and D) have no equivalent in the case of a grid-mesh partly saturated and partly unsaturated.

Nevertheless, N2(C) has the physical dimension of a squared BVF, and its expression closely follows, term by term, the physical logic explained in sections 6 and 7. It is thus our belief that it might be used in some applications, provided that the meaning of C is not overinterpreted.

Furthermore, by construction, the shape of this function is linked to the important issue already discussed of the change of sign of the terms in the second lines of (9) and (14). It is expected that the unstable (stable) feature of isentropic motions of moist air, which is in correspondence with saturated (unsaturated) conditions, must also have a neutral case in between. Even if the definition of a transition parameter remains a complex issue, well beyond the scope of the present work, N2(C) offers a monotonic path leading to some ‘moist neutrality’ for a value of C within the interval [0,1].

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