1. Introduction
a. Recent advances in the study of dry turbulence
The study of dry-air turbulence has made big progress in the past 13 years or so with recognitions in various scientific interests, namely (among others, less relevant here)
Formulations for surface drag coefficients and stability dependency functions [linking turbulent kinetic energy (TKE) and vertical exchange coefficients] can be made compatible across the whole range of possible stabilities (without critical Richardson number Ricr on the stable side), when using the equilibrium solution in the analytical derivation of the exchange coefficients of a system with prognostic TKE (Redelsperger et al. 2001, hereafter RMC01).
A more complex version of the pressure correlation formulations may be used (Launder et al. 1975) in the level-2.5 system of Mellor and Yamada (MY; Mellor 1973; Mellor and Yamada 1974, 1982) and still get a turbulence scheme of similar shape (Cheng et al. 2002, hereafter CCH02). For the comparison of both differing sets of hypotheses, see Galperin and Kantha (1989). An extension of CCH02 avoiding the appearance of a critical Richardson number in the more complex CCH02 system was later added (Canuto et al. 2008, hereafter CCHE08), albeit valid for stable stratifications only. Furthermore, the appendix of CCH02 indicates a path for linking with a level-3.0 closure for the leading term representing the conversion between turbulent kinetic and potential energies.
The absence of Ricr (i.e., the maintenance of turbulence in very stable stratifications) is explained via the velocity shear (Zilitinkevich et al. 2007, 2008) and the associated residual momentum flux. This corresponds to a sustained weak vertical diffusivity through the effect of internal waves in such conditions (Sukoriansky et al. 2005, hereafter SGS05). The behavior of stability dependency functions in stable conditions can then be described by the juxtaposition of a strongly turbulent and a weakly turbulent regime (Zilitinkevich et al. 2008) or equivalently a weakly stable, a transitional (not mentioned in the previous case), and a very stable regime (Sukoriansky and Galperin 2013).
The concept of stability dependency functions can still be used under some reasonable assumptions when prognostic aspects are encompassed not only in TKE but also in turbulent potential energy (TPE) or equivalently in the sum of both, turbulent total energy (TTE = TKE + TPE). Under the energy and flux budget turbulence closure theory (EFB) (Zilitinkevich et al. 2007, 2008, 2009, 2013, the latter hereafter ZEKRE13) a prognostic TKE–TPE(TTE) system is seen as the direct extension of the classical formulation (where only TKE is prognostic).
The most complete version of the EFB turbulence closure theory is based on budget equations for both turbulent energies (TKE and TPE) and for the turbulent fluxes of momentum and heat. Besides the usual downgradient transport term, the heat flux equation contains an additional term describing essentially positive (countergradient in stable stratification) contributions to the heat flux caused by temperature fluctuations. This additional flux, disregarded in conventional theories, is an inherent feature of the EFB theory ensuring no critical Richardson number. L’vov et al. (2008) also follow this approach for the heat flux equation.
For stable (or weakly unstable) stratifications, the Reynolds-averaging technique is not the only one leading to a compact formulation for both stability dependency functions (momentum SM and heat SH). A “wave range by wave range” stand-alone computation of the cumulated effects of anisotropy and of wave–turbulence interactions also delivers such functions (SGS05). This result is part of the wider quasi-normal scale elimination (QNSE) framework.
Although fundamentally distinct in the methodology from previous competing theories, QNSE data can be transformed from the wavenumber space to one of stability-representing entities (e.g., by using equilibrium assumptions). In addition, QNSE then confirms the absence of any critical Richardson number (Galperin et al. 2007). The asymptotic properties (in strongly stable stratifications) of the stability dependency functions, in agreement with the Osborn–Cox mixing model (Osborn 1980; Galperin and Sukoriansky 2010), that is,and of their ratio—the turbulent Prandtl number σt—σt ∝ Ri, all as obtained from the QNSE theory, are similar to the equivalent properties later derived in the EFB theory or in CCHE08.
For higher-order closure, there exists an heuristic relatively simple way to express third-order-moment (TOM) contributions to the second-order-moment (SOM) budgets (by simplifying the outcome of a complete reduction starting from the fourth order), with only three additional terms to consider (Canuto et al. 2007, hereafter CCH07). The link with the previous enumeration is that the general analytical solution (e.g., appendix of CCH02) for the “conversion term” is compatible with the CCH07 formulation. This allows applying the reduced complexity method (valid for the whole range of stabilities) to other systems than CCH02.
It transpires from the above that recent advances have been very rapid, in several partly interconnected directions, without major incompatibilities between them, but with no attempt (to our best knowledge) to bridge some gaps. This probably follows from some restricted conditions of application in most cases. The differing choices for the basic free parameters from one paper to the next might also be hampering the search for bridges between the formulations, although it should not matter, if equivalences are correctly expressed.
b. Aims of the paper
The aim of the present paper is thus to try and make some analysis toward the wider goal of a unique concept encompassing all the above (if ever possible) and to propose an intermediate analytical framework, aiming at “taking the best of each proposal while contradicting as little as possible the rest of it.” More precisely, we would like to do the following:
propose a solution valid for all stability conditions and practically compatible with numerical weather prediction (NWP) applications;
account for the impact of turbulence anisotropy on both momentum- and heat-related terms;
identify a suitable set of free parameters in our framework of stability dependency functions, in line with solutions proposed following the work of Schmidt and Schumann (1989, hereafter SS89);
solve the analytical problem of removing the critical Richardson number and find the most compact formulation;
emulate as much as feasible the QNSE and EFB systems on the basis of an adaptive ensemble consisting of MY-type analytical stability dependency formulations and of adjustable parameters and prepare a continuous extension of these emulations toward unstable stratification;
keep all this compatible with a TOM-type extension of the system, according to the CCH07 proposal; and
prepare a “natural evolution” of such a set of solutions with an additional prognostic equation for TPE(TTE), in the spirit of ZEKRE13.
We shall show below that these goals are all compatible. We therefore hope to convince the reader of the value of this approach to seek generality and modularity without giving up consistency.
In practice, we shall stick (implicitly or explicitly) to our closure-discretization method, which could be classified as somewhere between the level-2.0 and level-2.5 systems of MY. It uses stability dependency functions as in equilibrium conditions, making them dependent only on the Richardson number Ri, but relaxing this assumption for the evolution of TKE (see section 2), contrary to a more sophisticated use of equilibrium conditions in “quasi equilibrium” modeling (Galperin et al. 1988). For details about this difference, see section 8b.
Given its analytical orthogonal character with respect to the above, we shall treat the problem of the specification of the length scale only marginally. This and other aspects of the dry turbulent framework omitted here shall be addressed in a forthcoming paper.
Nonetheless, justifying the efforts described in the present paper only on the basis of the above arguments may still appear somewhat arbitrary. But there is a more important issue linked to such efforts. The extension from the dry-only case (i.e., even without water vapor) to the moist case (be it without or with saturation) destroys one of the basic pillars upon which all the above-mentioned “dry turbulence” theories are built, namely the double role of the potential temperature θ as a tracer of buoyancy and of entropy. This identity is lost as soon as water is present (because water vapor has a higher gas constant than dry air, because of latent heat release, or both). For NWP applications, it is a challenge to accurately predict both the partial cloud cover and the averaged buoyancy flux. This is a reason why, in our opinion, simple analytical solutions did not progress as much, in the past 13 years, as those for the dry case only. It is our belief that exploring separately the links between the moist case and all above-itemized “dry avenues of progress” would lead to a rather chaotic NWP situation. Hence we advocate first going through the above guidelines for streamlining the modern situation in the dry case, before going to the complex “moist” task with a minimum of modifications. The inference of the stability dependencies on the basis of the sole flux Richardson number Rif is at the heart of this preparation for a moist extension in our system. This extension shall be described in another forthcoming paper.
c. Constraints and scope of the work
With regard to NWP requirements, we note the Louis (1979, hereafter L79) scheme used in the Action de Recherche Petite Echelle Grande Echelle [ARPEGE; the global realization at Météo-France of the joint Integrated Forecast System (IFS)/ARPEGE common development with the European Centre for Medium-Range Weather Forecasts (ECMWF)] and Aire Limitée Adaptation Dynamique Développement International (ALADIN; the limited-area counterpart of the latter, developed and used by a community of 16 European National Meteorological Services) models. The last evolution of this scheme leads (J.-F. Geleyn et al. 2001, unpublished manuscript) to a formulation in terms of stability dependency functions having all the correct asymptotic behaviors (see above) required by our work plan. We could thus prepare, as an operational intermediate step and as a basis for further development, a “p-TKE” solution where the last version of the L79 scheme is complemented by advection and self-diffusion of a TKE value computed under the assumption that the level-2.0 basic Louis-type scheme should just be the stationary solution of the resulting system (Geleyn et al. 2006). On this basis, all proposals from the ensuing sections could be coded and pretested in a realistic NWP-type environment.
Owing to the restriction of our analysis to the hypothetic fully dry case, the results of such tests will not be presented here. The paper will only outline analytical developments and compare them with (laboratory and atmospheric) experimental, direct numerical simulations (DNS), and large-eddy simulation (LES) data and with the corresponding outcomes of the various schemes, which we try emulating and assembling at the same time.
The paper is organized in the following way. As “boundary conditions” for the central effort, section 2 describes our closure-discretization method and length-scale aspects and section 3 treats the problem of selecting the suitable set of free parameters and functional dependencies, which determine the whole set of model equations within constraints ensuring “No Ri(cr).” Section 4 explains the first development leading to our basic system and section 5 analyzes in more detail the surprising result that a second and completely independent path can lead to the same reduction of complexity for the main stability dependency functions of the CCH02 system. Sections 6 and 7 study the bridges between the resulting formulation and the stability dependency functions delivered respectively by the QNSE and EFB theories. Section 8 attempts to synthesize the various findings from a practical point of view. Section 9 briefly treats outlook aspects concerning higher-order terms (diagnostic and prognostic) and the extension to the moist case, while section 10 summarizes the findings. Appendix A lists the symbols and acronyms used in the paper. Appendix B lists the main relationships from the CCH02 paper frequently referred to in the main text. Appendix C gives analytical details about the section 5 complex derivation.
2. Model closure and discretization
This makes Eq. (5) in the time-continuous case identical to the prognostic TKE equation having source terms computed from Ri-dependent KM and KH in Eqs. (3) and (4) (like in a level-2.0 model) and from independent S2 and N2 (like in a level-2.5 model), hence the difficulty classifying our choice for closure and time discretization. The computation of ϕ3 for equilibrium conditions ensures a balance between its anisotropy part ϕQ (downgradient) and its TPE conversion part ϕconv (countergradient in stable stratification), which is one of the main factors for avoiding Ricr in our framework (see section 3). Given all these properties, we shall call this closure-discretization method “stability dependent adjustment for turbulent energy modeling.”
The above method is less sophisticated than that of a level-2.5 model but, contrary to the simpler level-2.0 case, it enables a fully consistent treatment of the prognostic TKE. This is an important aspect for present-day mesoscale NWP models with short time steps often reaching τϵ/(time step) > 1 (see Fig. 1). Indeed, autodiffusion and generation–dissipation of TKE cannot then be treated separately.
Histogram of ratios τε/(time step) in the planetary boundary layer after 7.5-h integration (time step = 90 s) of the ALADIN model on the CHMI research domain (2.2-km horizontal resolution, 87 vertical levels) with emulation extension of the QNSE scheme and L after Bougeault and Lacarrere (1989).
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
The stability of the above-described system is easy to control. Diffusion of momentum and heat is stable provided the (Deleersnijder et al. 2008, hereafter DHBD08) criterion for the corresponding level-2.0 model is satisfied (see section 8b). Stability of the implicit TKE solver is then ensured by diagonal dominance of its matrix, achieved by limiting the value of KE ⋅ τϵ above ⅜ of the square of the vertical local spacing.
We plan to generalize the above choice of length scale by adapting its computation to make it dependent on the (moist) squared BVF (Váňa et al. 2011). We also consider introducing memory when determining this generalized length scale by using a prognostic approach (following ZEKRE13, but replacing the time scale by the length scale as additional prognostic quantity).
3. Model framework
a. Free parameters
The following discussions rely on a combination of the RMC01 and CCH02 papers, complemented by a small key extract from CCHE08. This is indeed a sufficient starting point for the goals set in the introduction. Separately, the said papers cannot, however, cover all our needs. RMC01 does not account for the anisotropy of turbulence, while CCH02 falls short of eliminating the critical Richardson number in equilibrium conditions. Since CCH02 is the more complete of both papers, our notations (enumerated in appendix A) will mainly be taken from it, with additions either taken from RMC01 [Cϵ, CK, C3, C4, ϕ3(Ri)] or added for our purpose [P, R, Q, Oλ, ν, χ3(Ri)].
For the reader unfamiliar with the MY-type derivations of the CCH02 paper, the main relevant equations are reproduced in appendix B. Note that for the sake of simplicity, we transversally assume that the α3 coefficient of CCH02 is equal to zero [CCH02 authors also advocate such a step in the transition between their Eqs. (21b) and (21c)].
In RMC01 the basic set of free parameters Cϵ, CK, C3, and C4 [see Eq. (22)] is essentially the same as the one originating from SS89. We want to stay with four basic free parameters in our system but (i) choose them on targeted physical principles and (ii) get a clearer view of their interdependencies.
It is obvious from Eqs. (4) and (6) that C3, the physically important inverse of the Prandtl number at neutrality, has to be chosen as our third free parameter.
This introduces the constants RI and QI as one of the possible realizations of more general entities of our target system named R and Q (precisely defined at the end of this section and in sections 4 and 5). These variables play a key role for modulating the stability dependency under the influence of anisotropy.
The above relationship actually describes the balance between downgradient transport [first term in Eq. (20)] and TPE conversion [second term in Eq. (20)] by means of the ϕ3 decomposition. This balance resulting in ϕ3 going toward zero as inverse function of Ri for infinite stability is one of the main factors for having No Ri(cr) in turbulence schemes (Zilitinkevich et al. 2008).
The second required property for the absence of a Ricr is then χ3 going asymptotically toward a nonzero value for infinite stability. Provided both these requirements are met, the flux Richardson number asymptotically approaches to a nonzero value at infinite Ri, named the “critical flux Richardson number” and written Rifc.
Note that in our framework Rifc [determined from Eqs. (20) and (21) for Ri → ∞] is not an independent physical quantity but is given by the set ν, C3, and Oλ or by the set λ, F, and (γ1 ∧ C4) (see sections 4 and 5).
The above properties and thus No Ri(cr) are reached in the RMC01 formulation [via f(Ri) = χ3(Ri)(1 − Rif), with their f function used as a filter ensuring the equilibrium condition], but this outcome is somewhat biased by their enforcement of isotropy (R ≡ 1 and Q ≡ 1). The correct asymptotic behaviors also appear (in a more straightforward way) in the QNSE and EFB frameworks, and we shall try to make them general by enforcing their appearance in all our CCH02-related derivations.
Looking back at CCH02, there remains one value, which is independent from the quartet (Cϵ, ν, C3, Oλ): namely, the parameter λ4 controlling the direct influence of the heat flux on the momentum flux [see Eq. (B9)]. Implicitly, λ4 is set to zero in the RMC01 system. This is one of the main factors for absence of Ricr in RMC01 and it inspired the derivation of model I in our framework (see section 4). But the hypothesis λ4 = 0 is rather restrictive (e.g., Shih and Shabbir 1992; Canuto 1994). This issue will be extensively discussed in sections 4 and 5 (in rather contrasted terms). In the analytical path developed in section 5, we shall keep λ4 to its CCH02 formulation [first part of Eq. (B28)].
Concerning orders of magnitude, the basic recommended setup of CCH02 (λ = 0.4, F = 0.64, γ1 = ⅓, and C4 = 2) leads here to ν = 0.526, C3 = 1.18, and Oλ = 0.667, which are all physically rather reasonable values.
b. Core equations
Logically the variables P, R, and Q are determined by the sets of parameters ν, C3, and Oλ or by the set λ, F, and (γ1 ∧ C4). Depending on the characteristics of the scheme that we want either to make more compact (CCH02) or to emulate (QNSE or EFB), they may be either constants or stability dependent functions. Table 1 gives an anticipation of what we shall find in the next four sections for this key aspect. Our proposed scheme is compactly written, but it adapts itself in this way to various and rather differing situations.
Properties (Const: Constant; Ri fun.: function of Ri) of P, R, and Q variables for the schemes presented below. Model I and model II: modified CCH02 scheme (see sections 4 and 5); eeQNSE: emulation extension of QNSE (see section 6); eeEFB: emulation extension of EFB (see section 7).
This equation will be valid for all realizations of CCH02 and will become an excellent way to fit the stability dependency functions in our emulation extension of QNSE.
4. Equations: The basic system
In this section, we develop a modification of the CCH02 system by setting λ4 to zero, thus canceling the direct influence of the heat flux on the momentum flux. Such an approach leads to a simplified scheme without Ricr, which we shall name model I. Because of the rough nature of the simplification, we do not recommend the use of model I in practical applications, but it still represents a basic element in our “family” of models that avoid Ricr by construction. Furthermore, we shall see later that it is an important step in the emulation extension of the EFB system.
As written in section 3a [Eqs. (12)–(15)], RI and QI can be directly computed from λ and F or from ν and C3. As seen from Eq. (45), PI, which in the case of model I is equal to Rifc, additionally depends on the combination of the coefficients C4 and γ1 or, equivalently, on the joint parameter Oλ.
The compact set of equations for model I [Eqs. (41), (47), (48), and (44)] corresponds to the anticipated equations [Eqs. (28)–(31)]. This characteristic is linked to the absence of influence of the vertical heat flux on the vertical momentum flux via the partly arbitrary assumption λ4 = 0, in disagreement, for instance, with experimental data from Shih and Shabbir (1992) where β5 is surely nonzero. However, such an approximation also appears in RMC01, inspiring our model I derivation. In the more complex EFB scheme, the vertical heat flux also has no influence on the vertical momentum flux [see Eq. (34) in ZEKRE13], but it is then associated with a specific set of relationships for the TKE components. This discrepancy in model I with respect to the more complete EFB solution can be corrected by introducing a variation with respect to stability of the otherwise constant (see Table 1) variables P and R (see section 7).
Note also that assuming λ4 = 0 in model I yields an unrealistic isotropic Az = ⅓ value in a convective flow without shear [see Eq. (39)]. However, in our basic closure-discretization method, the determination of TKE components has a purely diagnostic purpose. The direct evaluation of Az influences the computation of turbulent fluxes only when we go to an extension with TOMs parameterization (see section 9). In such a case, the anisotropy properties of convective flows should automatically be improved owing to the contribution of plume-type TOMs terms.
5. Equations: A system with higher initial complexity
This indeed leads to a system without Ricr, but with two drawbacks: (i) the system becomes more complicated and (ii) it is then only valid for the stable case.
Concerning this choice, it could be argued that the coefficients of the basic Reynolds decomposition system should ideally be constants (Lewellen 1977). But in the same work (p. 247) a model is still considered valid, if such coefficients vary, provided it is on the sole basis of dimensionless parameters, such as gradient Richardson number Ri (or, by extension, σt). A similar approach was also used since the CCHE08 publication in Kantha and Carniel (2009) and in Kitamura (2010).
This result is surprising (since obtained for constant values of R and P on both sides) and simpler than the one of CCHE08 while also valid for the unstable range, even if derivation is rather tailored to our closure discretization.
Values of QII for model II.
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
Other differences between models I and II are less obvious, owing to the identical shape of the stability-dependency functions χ3 and ϕ3. While C3 and ν indeed depend on λ and F in the same way in both systems [see Eqs. (12) and (14)], the dependency of constant R and P = Rifc on λ, F, and Oλ differ [see Eqs. (43) and (45) in comparison with Eqs. (62) and (63)].
6. Emulation of QNSE: Bridge with the outcome of a spectral analysis of turbulence and waves
QNSE is a theory of turbulence based on a spectral analysis of the flow that uses a radically differing approach from Reynolds stress modeling (RSM). QNSE delivers general spectral-type results about turbulence activity, which may be converted to a Reynolds-averaging Navier–Stokes (RANS) framework with the help of some linking hypotheses between scales and Richardson numbers [Eqs. (182)–(189) of SGS05]. In such a case, one directly gets “data” allowing linking of χ3 and ϕ3 (separately) with Ri. The challenge is to see whether the specific framework built above [RSM on the basis of CCH02 and RMC01; Eqs. (28)–(31)] can be used for emulating QNSE data in stable stratification and possibly extending such an emulation (for practical applications) toward unstable stratification.
We first set C3 = 1.39 and Rifc = C3/(C3 + 2.3), 2.3 being the opposite of the derivative of ϕ3 with respect to Ri at neutrality, all this according to QNSE publications.
We can then verify that the separate datasets of χ3 and ϕ3 in QNSE seem to agree (cf. the second and third columns of Table 2) with the equation linking ϕ3 and χ3 in our framework [Eq. (35)] when we assume a constant P = Rifc. This enables us to fit only one of χ3 and ϕ3 and to keep consistency between them. At first sight, it would seem simpler to deduce χ3 from ϕ3 than the reverse. But we chose to start with χ3 owing to the difficulties of assessing ϕ3 at high Ri values [data only approximately correct and need to somewhat tune them down—progressively with increasing Ri; see section 5.1 of Sukoriansky et al. (2006)].
Comparison of QNSE data with their emulation. Read 
We still have to verify whether this way of retrieving ϕ3 gives a good quality fit with respect to the corresponding corrected QNSE data. Going back to Table 2 and comparing this time the second and fifth columns, we see that the quality of the ϕ3 fit nearly matches the very good one for χ3 (first and fourth columns), as a confirmation of the validity of this approach.
As in the stable case, we compute in unstable stratification ϕ3 from 
The ϕ3 and χ3 functions having finite asymptotic values for the very unstable case, one indeed gets FH correctly proportional to (−Ri)1/2.
The main difference between eeQNSE and models I and II is the stability dependent R in eeQNSE. Even though ReeQNSE varies with stability (see Fig. 3), it is mostly ±10% around a 0.4 value, supporting our attempt to emulate and extend the QNSE behavior on the basis of our framework.
Implicit values of R computed from χ3 in eeQNSE [see Eqs. (67) and (68)].
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
To complement eeQNSE with values for the free parameters ν and Oλ, which are not delivered by the QNSE framework, we use (for such purpose only) a constant R = R∞ = 0.44 and assume that the internal links between ν, C3, and Oλ and R and P in eeQNSE are the same ones as in model I or II [see Eqs. (12), (14), (43), (45), (62), and (63)].
The QNSE theory delivers no decomposition of ϕ3 into its anisotropy (ϕQ) and conversion parts. We may however achieve this decomposition in eeQNSE on the basis of parameters computed from a constant 
7. Emulation of EFB: Generalization of the framework
EFB is an RSM scheme based on the budget equations for TKE and TPE and for turbulent fluxes. Consideration of the TPE conversion term in the heat flux budget and an appropriate stability dependent parameterization of TKE components ensure the absence of Ricr in the EFB scheme. The aim of this section is to compare models I and II with the “minimal prognostic model” variant of the EFB scheme (the one corresponding best to our closure-discretization choices) and to explore whether we may put both realizations under a common hat.
Until now EFB appears very similar to the models I and II with different values of ν, C3, and Oλ. Let us now look at the shapes of stability dependency functions characterized by the variables P, R, and Q.
Fit of REFB in Eq. (86) and indirect fit of PEFB [via Eq. (82)] compared to implicit R and P in EFB and 
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
The differences between EFB and models I and II in R, P, and Q are connected to a different parameterization of the pressure correlation terms. While EFB decouples the momentum flux from the heat flux [in pressure correlation term for momentum; Eq. (34) in ZEKRE13] in an equivalent way to model I (λ4 = 0), pressure correlation terms in the budgets for TKE components are different. Namely, EFB proposes a partly independent model following results from simulations and atmospheric data measurements [Eqs. (48a)–(48c) in ZEKRE13].
The above properties of R, P, and Q mean that links between ν, C3, Oλ and P, R, Q in models I and II are not valid in the EFB scheme. The specific emulation of EFB within our framework must take this into account by directly fitting one of the stability dependencies while preserving its analytical links with the other ones.
The quality of the fit of REFB and of the indirect fit of PEFB [via Eq. (82)] can be assessed by going back to Fig. 4.
As soon as we know both P and R, we can compute Rif from Eq. (31) and this delivers then indirect fits of χ3 and ϕ3 (the quality of such fits can be assessed from Fig. 5).
Quality of the indirect fit [computed from Eqs. (86), (82), and (31)] for the χ3 and ϕ3 functions in EFB.
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
8. Practical aspects
a. Numerical realizations and intercomparison of schemes
Comparison of our models I and II with QNSE and EFB schemes showed that there exists a way of integrating all four occurrences (or very good approximations of them) in one single more general framework (three free parameters ν, C3, and Oλ and three variables P, R, and Q).
Table 3 shows a comparison of numerical values of the schemes with activated links (internal or taken over from models I and II in case of eeQNSE).
Values of free parameters ν, C3, Oλ, and Cε and variables P, R, and Q for different schemes: basic model I and II—modified CCH02 scheme with λ = 0.4, F = 0.64, and Oλ = 2/3 (see sections 4 and 5); tuned model II—model II with λ = 0.4, F = 0.64, and Oλ = 0.29; eeQNSE—emulation extension of QNSE (see section 6); eeEFB—emulation extension of EFB (see section 7).
All schemes are relatively close to each other concerning C3 and especially ν, but they differ more in Oλ (lower values especially for eeEFB, but even for eeQNSE), which has an indirect influence on Rifc via Eq. (45) or (63). Models I and II are, however, not restricted to basic setups of the Oλ value and can be tuned toward eeQNSE or eeEFB as anticipated in the third column of Table 3. Such targeted realizations cannot provide stability dependent R(Ri) and/or P(Ri) values (like in eeQNSE and eeEFB), but they represent possible model I and model II setups with via Oλ modified intensity of the TPE ⇔ TKE conversion.
A comparison of models (adjusted model II, basic models I and II, eeQNSE, eeEFB, CCH02, and CCHE08) is displayed in Figs. 6–8 together with smoothed scatterplots generated from various available sources of verification data, using the same choices as CCHE08 and/or ZEKRE13. The results show that the adjustment of Oλ in the tuned model II significantly improves the anisotropy properties, especially concerning the momentum flux. For the vertical TKE component, there is a clear split between the eeEFB specific choice (see section 7) on the one side (a bit closer to the bulk of the verifying data) and the rest of the choices on the other side (among which the tuned model II behaves relatively well). The results for the heat flux confirm the already clearly seen problems against any basic use of model I and the need of tuning Oλ in model II, as a complement to using the basic relationship in Eq. (52).
Scaled (with values at Ri = 0) dimensionless squared momentum flux—
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
As in Fig. 6, but for the relative vertical component of TKE–Az and with data added from DNS (Stretch et al. 2001) and from meteorological observations (Engelbart et al. 2000).
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
As in Fig. 6, but for scaled (with values at Ri = 0) dimensionless squared heat flux—
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
The differences between the various solutions can also be visualized in the 3D space of R∞, P∞ = Rifc, and C3 (the variables that directly influence the computation of exchange coefficients KM and KH). In Fig. 9, the eeQNSE and eeEFB single points are outside the basic model I and II isosurfaces but are closer to the adjusted Oλ = 0.29 isosurface on which lays the tuned model II point.
Points for basic models I and II (λ = 0.4, F = 0.64, Oλ = 2/3) for tuned model II (λ = 0.4, F = 0.64, Oλ = 0.29), for eeQNSE (R∞ = 0.44, P∞ = 0.377, C3 = 1.39), and for eeEFB (R∞ = 0.282, P∞ = 0.25, C3 = 1.25) and Oλ isosurfaces for models I and II (
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
A comparison of the resulting stability dependency functions ϕ3, χ3, and also of Rif(Ri) is displayed, for five cases (the basic setups of models I and II, tuned model II, eeQNSE, and eeEFB) in Figs. 10–12. In the latter two, the superposition of the eeQNSE and eeEFB curves in the unstable case is the result of a pure coincidence 
Stability dependency function for basic model I and II setups, tuned model II, emulation extension of QNSE, and emulation extension of EFB (set text) for (top) the whole range of Ri and (bottom) stable stratification.
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
As in Fig. 10, but for the stability dependency function.
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
As in Fig. 10, but for the dependency Rif(Ri) for (top) unstable stratification and (bottom) stable stratification.
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
The most significant difference between models I and II can of course be seen on the internal split of ϕ3 into ϕQ and ϕconv (for this, see Fig. 13). It is clear that eeEFB has the same asymptotic behavior as a model I; in the eeQNSE case the split of ϕ3 can equally well happen in model I or model II mode.
As in Fig. 10, but for the anisotropy stability dependency function for the whole range of Ri. The eeQNSE function has two modes (in model I and in model II).
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
b. Realizability and stability analysis
Owing to the targeted binding of potentially independent N2 and S2 stability parameters into Ri (or Rif) via the equilibrium TKE equation (see section 2), our closure-discretization method ensures not only the absence of Ricr but also always getting physically sound solutions in numerical simulation conditions.
As seen from Fig. 14, all chosen model realizations systematically have positive real parts of 
As in Fig. 10, but for the real parts of the eigenvalues of matrix 
Citation: Journal of the Atmospheric Sciences 71, 8; 10.1175/JAS-D-13-0203.1
More sophisticated models influenced by independent N2 and S2 increase the theoretical accuracy of the parameterization, but at the price of needing protection against the risk of unphysical results in extreme cases. The “quasi equilibrium” model requires a bounding of the GH value and the original level 2.5 of MY additionally requires a protection to be applied to the positive GM value.
Even if level-2.5 model closure is out of the main scope of our paper, it is possible to use Eqs. (37)–(38) and (55)–(57) as basis for level-2.5 closures of model I and model II, respectively. Both such potential schemes are suitable realizations at level 2.5 since they are unaffected by any TKE equilibrium approximation.
Both schemes are stable according to the stability criterion given by DHBD08 for a level-2.5 closure (see section 3.2 in DHBD08) provided GM and GH are adequately limited (see above). In case of model I, only an upper limitation of GM by a 134.0 value is required. For the basic and tuned (Oλ = 0.29) setups of model II, it is necessary not only to limit GM by the same maximum value of 134.0 but also to limit GH by minimal respective values of −10.8 and −26.0.
9. Outlook toward higher order terms and moist turbulence
The variety of the stability dependencies from one case to the next shown in the previous section indicates that our common framework, far from forcing an artificial convergence of the solutions proposed in the literature, just makes their intercomparison as unbiased as possible. Such comparisons may touch, for example, the L specification, the choice of Cε, the extensions toward diagnostic TOMs and/or prognostic TPE, and the consideration of moist turbulent aspects. Conversely, such extensions, in principle necessary for the best possible use in NWP applications, can be derived from one of the considered theories and then safely translated to any emulation or simplification of the other ones. This will be briefly outlined in this section.
The time-stepping organization then looks as follows. In the p-TKE spirit, Eq. (2) is solved first, but with flux values from the previous time step used to replace the handling of Eqs. (3) and (4) by a direct computation of the shear and buoyancy terms. Such a choice implicitly solves the problem of the incorporation of the TOMs contributions into the conversion term. A prognostic equation for TTE (from which the buoyancy term vanishes) is solved in a parallel way. The new values of TKE+, TTE+, and Π+ are then used to compute the vertical exchange coefficients KM and KH. Those are then used for independent computations of the momentum and heat fluxes. In the latter case, factorization of the whole Eq. (91) by KH is similar to the EFB corresponding level-3.0 proposal—the crucial ingredient of the algorithm.
The extension to the moist case can best be considered with the two previous extensions already granted. In such a case, apart from the classical “Betts’ transform” change of diffused heat and moisture variables, there remain three problems: how to compute the buoyancy flux in case the cloud cover is neither zero nor one, how to define an equivalent of TPE where the total moisture plays a symmetric role to that of the heat variable, and how to modify KH, assumed to be the same for heat and for moisture? The work reported here does not help for the first and the third issues. As all our stability dependencies can be expressed in terms of Rif (and hence if needed of Π) we are able to reduce the solution of the second problem to that of the first one. The formulation is, however, rather complex and hence out of the scope of the present paper.
10. Conclusions
We showed that, despite their sometimes widely diverging basic hypotheses, the dry turbulent schemes described in the CCH02 MY-type solution, in the QNSE theory, and in the relevant configuration of the EFB approach can all be adapted to our equilibrium-oriented closure-discretization method. The resulting compact and transversal framework [Eqs. (28)–(31) and Tables 1 and 3] matches each of the schemes’ basic behavior but helps to better understand their respective “signatures.” Essential properties of the new framework are (i) validity for all stability conditions with absence of Ricr and (ii) accounting for turbulence anisotropy in both momentum- and heat-related terms. The RMC01 paper inspired our discretization, as well as the related choice for the length-scale specification.
In the CCH02 case, as already identified by the same authors in CCHE08, some treatment was needed to obtain a No Ri(cr) behavior. On the basis of the slight simplification α3 = 0, we found out that strictly following (i.e., without the initial approximation advocated there) the CCHE08-proposed path delivers the same stability-dependency functions as an unrelated method directly inspired by RMC01 (but with anisotropy effects duly accounted). This surprising result can hardly be the fruit of a coincidence. We believe that our resulting system indeed possesses some basic properties that must be easily transferable to any set of equations valid for the whole range of stabilities and leading to both a behavior of the No Ri(cr) type and a separation between the χ3 and ϕ3 dependencies on Ri (or Rif).
This belief is reinforced by the fact that the obtained system, once codified in its more general shape (i.e., with three free parameters and three variables determining the shape of stability dependency functions), is able to accurately fit the relevant functional dependencies derived in the independent QNSE and EFB theories for the range of positive Ri values. Here no simplification or analytical manipulation is needed, just a consistent fitting procedure, on the basis of only one of the truly independent entities in the QNSE case, as well as in the EFB case. The strength of this approach is demonstrated (in both cases) by the quality of the indirect fit(s) of the other entities.
Despite the sometimes complex nature of the derivations leading to it, our scheme remains analytically very simple. This helps devising, with a minimum of arbitrariness, extensions to the unstable case, which neither QNSE nor EFB proposals possess in their original versions.
Identity and simplicity of the analytic shapes does not, however, mean uniformity of the resulting functional dependencies. On the contrary, each of the derived or emulated extended solution possesses its own way to condition the setup of the general shape of our formulation. Comparison between the values of free parameters (Table 3) and of stability dependency functions shapes shows for instance the signature of the wave–turbulence aspects of QNSE (Fig. 9) or indicates the impact of an independent model describing the anisotropy of the turbulent flow in EFB (Fig. 13).
We studied in detail five cases of our framework realization (basic model I and II, tuned model II, eeQNSE, and eeEFB). The comparison with experimental, DNS, and LES data shows the flexibility of our framework. The DHBD08 analysis confirms that these realizations of our framework are numerically stable.
The above five cases represent a progressive gradation (first Q, then R, and finally P) in transitions of variables from constant-value status to dependency on Rif. This is why we are confident in the representativeness of the present study’s outcome. However, we do not claim to have explored all issues related to this “codifying effort.”
Finally, owing to its full compatibility with the solutions proposed in CCH07 and ZEKRE13, our simplified framework can be extended without additional constraints both to the complementing of the conversion term by nonlocal TOM terms and to the prognostic handling of TTE or to their combination. The path toward a full moist turbulence extension is not as clear cut, since all that we reported here is, for instance, orthogonal to the shallow-convection closure problem. But the fact of being able to express all stability dependencies in function of the flux Richardson number is a nice asset for the formulation of moist source–sink terms of TKE. The fact that our set of equations systematically covers the whole range of possible stabilities might be here a big advantage, owing to the push toward instability created by the water vapor–induced change of density and/or by the latent heat release.
Acknowledgments
The authors thank J.-L. Redelsperger for the initial incentive to put in relation the RMC01 and CCH02 papers. They are grateful to S. Zilitinkevich and B. Galperin for their help in past years for better grasping the relevant aspects of the EFB and QNSE theories. They are most indebted to B. Galperin and to both other referees who considerably helped enhance the relevance of the paper through their pertinent questioning and via positive reorientation suggestions. Over several years, constructive discussions with P. Unden, D. Mironov, P. Marquet, E. Bazile, N. Pristov, M. Gera, R. Brožková, J. Cedilnik, E. Machulskaya, and M. Tolstykh contributed to the shaping of this paper. G. Carver skilfully helped to improve the wording of the manuscript. Support of the ALADIN community, especially in its RC-LACE component (ALARO physics project), was essential for the completion of the work, together with the support of the Czech Grant GACR No P209-11-2405. Part of this study was also inspired by the aims of the EU-funded COST ES0905 Action.
APPENDIX A
List of Symbols and Acronyms
(⋯)′ Fluctuation: 
(⋯)I Model I
(⋯)II Model II
(⋯)0 Value at Ri = 0
(⋯)∞ Value for Ri → ∞
Az Ratio of vertical component of TKE to TKE itself
α3 Coefficient determining λ6 and λ7 (see appendix B)
β5 Coefficient controlling the direct influence of heat flux on momentum flux (see appendix B)
C1–2 Coefficients in TOM terms
Cε Free parameter, linked to the choice of L and controlling the intensity of turbulent dissipation
CK Coefficient: 
Γ Coefficient of the Osborn–Cox model
γ1 Coefficient controlling the influence of TPE on heat flux (see appendix B)
d1–5 Coefficients in CCH02 (see appendix B)
D Denominator in SM and SH (see appendix B)
DNS Direct numerical simulations
EFB Energy and flux budget
F Coefficient influencing mean shear–turbulence interactions for pressure correlations (see appendix B)
FH Stability dependency function for heat in L79: KH = lmlhSFH, where lm,h represents Prandtl-type mixing lengths
f(Ri) =χ3(Ri)(1 − Rif)—function used as a filter ensuring the equilibrium condition in RMC01
GM =τ2S2
GH =τ2N2 = RiGM
θ Potential temperature
KE Vertical turbulent exchange coefficient for TKE: 
KM Vertical turbulent exchange coefficient for momentum (see also SM): 
KH Vertical turbulent exchange coefficient for the thermodynamic quantity (see also SH): 
κ Von Kármán constant
L Length scale in the prognostic TKE formalism
LES Large-eddy simulation
lm Mixing length from similarity laws
λ4 Parameter controlling the direct influence of the heat flux on the momentum flux [see Eq. (B9)]
N Brunt–Väisälä frequency (BVF)
NWP Numerical weather prediction
ν =(CεCK)1/4 = (2SM,0)1/4; free parameter (controlling the overall intensity of turbulence)
Oλ 
P Variable describing the joint effect of flow’s anisotropy and TPE conversion on the turbulent heat exchange with 
Q Variable describing the effect of the flow’s anisotropy on the turbulent heat exchange
QNSE Quasi-normal scale elimination
R Variable describing the effect of the flow’s anisotropy on the turbulent momentum exchange
RANS Reynolds-averaging Navier–Stokes
Ri Gradient Richardson number
Ricr Critical Richardson number
RSM Reynolds stress modeling
SOM Second-order moment
SM Stability dependency function for (u, υ): KM = eτSM
SH Stability dependency function for θ: KH = eτSH
s0–6 Coefficients in CCH02 (see appendix B)
TKE =e; turbulent kinetic energy
TOM Third-order moment
TTE =TPE + TKE; turbulent total energy
Th Stability dependency function in TOM terms
τθ Temperature variance dissipation time scale [see Eq. (B8)]
τp,θ Pressure–temperature correlation time scale
τp,υ Pressure–velocity correlation time scale
u ≡u1; zonal wind component
υ ≡u2; meridional wind component
w ≡u3; vertical wind component
ϕconv 
ϕQ Anisotropy part of ϕ3 stability dependency function
APPENDIX B
Relationships in CCH02 Paper
This appendix lists the main relationships from CCH02 frequently referred to in the main text, arranged by type of equation.
a. Heat flux
b. Heat flux by neglecting 
c. Temperature variance equation–TPE equation
d. Momentum flux
e. Stability dependency functions
f. TKE equation in equilibrium condition
APPENDIX C
Derivation of Model II
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