## 1. Introduction

Reynolds stress turbulence modeling began in the early 1940s (Chou 1940, 1945) and since then it has been developed by both physicists and engineers (e.g., Rotta 1951; Lumley and Khajeh-Nouri 1974; Launder et al. 1975; Pope 1975; Zeman and Lumley 1979; Speziale 1991; Shih and Shabbir 1992). The parameterizations of the turbulence closures have been formulated theoretically, verified experimentally (including comparison with the ever more reliable LES data), and applied to various engineering flows. In the geophysical applications, Mellor (1973), Mellor and Yamada (1974), and Mellor and Yamada (1982) pioneered the use of turbulence closure models to study the planetary boundary layer (PBL). The Mellor–Yamada (MY) model and its numerous variants have been more successful in the simulation of the PBL than many of the empirical models and have been widely used to describe the atmospheric PBL and the oceanic mixed layer. The MY models are, however, not without deficiencies. Comparison of MY model results with measured data and LES data show consistent discrepancies, and close examination indicates that the weakness of the model comes from three sources: 1) a crude parameterization for the pressure–velocity and the pressure–temperature correlations, 2) the use of a single “master” length scale (all the length scales corresponding to different processes are assumed to be proportional to a master scale), and 3) a downgradient approximation for the third-order turbulent moments. These three aspects can be handled as three independent components in the model development and each of them deserves a separate discussion. Along with many other efforts, the present authors tried to address items 2 and 3 elsewhere (Cheng and Canuto 1994; Canuto et al. 1994, 2001). The present paper concentrates on item 1, that is, how to improve the parameterization for the pressure correlations, thus generalizing the MY models and improving the comparison with both measured and LES data. The LES has been widely and successful employed in the PBL (e.g., Moeng and Wyngaard 1986, 1989; Moeng and Sullivan 1994) and LES results have been regarded as experimental data, which are useful to guide and to test theoretical studies.

Let us look at deficiency 1 of the MY models and its variants (e.g., Galperin et al. 1988) more closely. First, the level 2 (see section 5 for details) of these models predicts too low a critical Richardson (Ri_{c}) number (around 0.2), beyond which the turbulence ceases to exist, while both measurements and LES data (e.g., Webster 1964; Young 1975; Wang et al. 1996) indicate that the critical value is around unity. Second, when applied to the neutral boundary layer, assuming production equals dissipation, none of these models is capable of differentiating between the vertical and lateral components of the turbulent kinetic energy, *w*^{2}*υ*^{2}

As we will show below, these deficiencies are associated with the oversimplification of the parameterizations of the pressure–velocity correlation Π_{ij} and pressure–temperature correlation ^{θ}_{i}_{ij} and ^{θ}_{i}_{ij} and ^{θ}_{i}*S*_{ij}, and the vorticity tensor *R*_{ij}, as well as buoyancy terms related to the heat fluxes. In addition, the rapid part of ^{θ}_{i}*θ*^{2}_{ij} include only the slow part and some of the rapid part (the term proportional to *eS*_{ij}, where *e* is the turbulent kinetic energy); for ^{θ}_{i}

In sections 2, 3, and 4, we introduce the basic equations and the new turbulence closure. In section 5, we derive the new algebraic Reynolds stress and heat flux model for the PBL. The new model is presented in three different “levels” according to MY's terminology. In section 6, a new value of the critical Richardson number is derived and discussed. Model constants are determined in section 7. In section 8, we compare the new model and the MY model with measured and LES data, where we can see that the new model matches the measured and LES data better than previous models. Conclusions are presented in section 9.

## 2. Mean field equations

To model a PBL, we need both mean and turbulent variables. The governing equations for mean fields are as follows:

*U*

_{i}:

*u*

_{i}is the

*i*th component of the turbulent velocity fluctuation,

*g*

_{i}= (0, 0,

*g*) is the gravitational acceleration,

*P*is the mean pressure,

*ρ*is the mean density, Ω

_{j}is the rotation of the earth,

*τ*

_{ij}is the Reynolds stress, and

*h*

_{i}is the heat flux.

*U*

_{g}and

*V*

_{g}as follows:and the rotation term can be approximated as

*ϵ*

_{ijk}

_{j}

*U*

_{k}

*f*

_{c}

*ϵ*

_{ij3}

*U*

_{j}

*x,*

*y,*and

*z*are the eastward, northward, and vertical directions, respectively,

*f*

_{c}= 2Ω sin

*ϕ*is the Coriolis parameter with Ω the angular velocity of the earth and

*ϕ*the latitude. In Eq. (1b), the horizontal temperature gradient can be approximated with the thermal wind relation,where

*α*is the volume expansion coefficient.

*U*and

*V*and for the mean potential temperature Θ in the PBL can then be written as

## 3. Turbulence equations

*τ*

_{ij}:where

Here, Π_{ij} is the pressure–velocity correlation tensor, *ν* is the molecular viscosity, *ϵ* is the dissipation rate of the turbulent kinetic energy *e,* and *D*_{ij} is the diffusion term.

*e*:

*h*

_{i}:wherewhere

^{θ}

_{i}

*D*

^{h}

_{i}

*h*

_{i}.

*θ*

^{2}

*χ*is the molecular conductivity,

*D*

_{θ}is the diffusion of the temperature variance, and

*ϵ*

_{θ}is the temperature variance dissipation rate.

In the present study, terms containing the molecular viscosity *ν* and molecular conductivity *χ* have been neglected, except for *ϵ*_{ij} and *ϵ*_{θ}. In addition, in the second-moment equations, rotation has also been neglected. The modeling of the third-order moments exceeds the scope of the present paper, but the interested readers may refer to recent work on the subject (Canuto et al. 1994, 2001). As already stated, in this paper we concentrate on the closure parameterization of the correlations Π_{ij} and ^{θ}_{i}

## 4. Turbulence closure

### a. ϵ and ϵ_{θ}

*ϵ,*

*ϵ*

_{θ}

*e*and

*θ*

^{2}

*e*and

*θ*

^{2}

*ϵ*and

*ϵ*

_{θ}can and have been derived, they are of little practical use since most of the terms are difficult to interpret physically, and thus difficult to represent. A phenomenological equation for

*ϵ*has been proposed long ago and in the case of pure shear flows, it has been used quite extensively in spite of containing two adjustable coefficients. When buoyancy is included, the number of unknown coefficients increases and it is difficult to calibrate them so as to assure any type of generality. Due to these difficulties, it is customary to use an alternative approach, namely one employs the basic relation (Batchelor 1971)where Λ is the dissipation length scale. Rewriting (5b) in terms of Λ and

*τ*defined asone hasIn section 4b we will discuss how

*τ*is determined, thus the determination of

*ϵ*reduces to the modeling of Λ, a variable that has been extensively studied in the past. Here, we employ the Blackadar–Deardorff (Blackadar 1962; Deardorff 1980) model wherebywhere

*B*

_{1}is a constant, which will be determined in section 7. Equation (5e) has been used in the PBL context by many authors (e.g., André et al. 1978; Hassid and Galperin 1983; Galperin et al. 1988).

*ϵ*

_{θ}is even more difficult to calibrate than the

*ϵ*equation and also too complicated to use. We use instead the parameterizationand we will determine the timescale

*τ*

_{θ}in section 7.

### b. e and θ^{2}

^{2}

*e*and

*θ*

^{2}

*θ*

^{2}

*e*equation by assuming that production of

*e*equals its dissipation:

*P*

_{b}

*P*

_{s}

*ϵ,*

*P*

_{s,b}represent the production terms due to shear and buoyancy, respectively. Each production term is proportional to the gradient of the mean variable in question times a turbulent diffusivity

*K*[e.g., see Eqs. (16a,b)]. In terms of the basic variables

*e*and

*ϵ,*

*K*has the dependenceand thus the solution of (5g) does not yield

*e*or

*ϵ*separately but only their ratio

*τ,*

*τN*

*f*

*f*(Ri) is thus uniquely determined by (5g) as a function of Ri and, as expected,

*τ*grows with Ri indicating that for a fixed shear, the larger the stratification, the weaker is the turbulence and the longer is the timescale

*τ.*In the limit

*τ*→ ∞, one may consider the flow to have become almost laminar, namely the eddies have an infinite lifetime, they no longer break up, thus no cascade process exists. On the other hand, for small Ri, the stratification is weak, turbulence dominates, and the eddies tend to break up quite easily due to the strong nonlinear interactions. In other words,

*τN*≫ 1 corresponds to weak turbulence while

*τN*≪ 1 corresponds to strong turbulence.

### c. Pressure correlations

_{ij}and

^{θ}

_{i}

_{ij}and

^{θ}

_{i}

*b*

_{ij}is the traceless Reynolds stress tensor defined as follows:The other tensors are defined as follows:where

*S*

_{ij}and

*R*

_{ij}are shear and vorticity, respectively.

_{ij}, the MY (1982; Mellor 1973) and Kantha and Clayson (1994) models only consider:for

^{θ}

_{i}

^{θ(1)}

_{i}

*τ*

^{−1}

_{pθ}

*h*

_{i}

^{θ(2)}

_{i}

^{θ(3)}

_{i}

^{(2)}

_{ij}

^{θ}

_{i}

^{θ(2)}

_{i}

^{θ(1)}

_{i}

*τ*

^{−1}

_{pθ}

*h*

_{i}

^{θ(2)}

_{i}

^{θ(3)}

_{i}

*γ*

_{1}

*β*

_{i}

*θ*

^{2}

_{ij}and

^{θ}

_{i}

*α*

_{1}and

*α*

_{2}were either taken to be identical or set to zero in Eq. (6b), and

*S*

_{ij}and

*R*

_{ij}were set to have the same coefficient in Eq. (6c).

In section 5, we will derive new expressions for the stability functions (*S*_{M} and *S*_{H}) using the more complete pressure correlations. In the following sections, we will show that previous parameterizations of the pressure correlations are at the root of some model deficiencies, for example, the inability to match the data in the neutral boundary layer as well as in the stably and unstably stratified flows.

## 5. Algebraic Reynolds stress and heat flux models

### a. Algebraic equations for the second moments

*b*

_{ij}, defined in Eq. (6d):where

_{ij}, one obtains the following algebraic equation for

*b*

_{ij}

*b*

_{ij}

*λ*

_{1}

*eτS*

_{ij}

*λ*

_{2}

*τ*

_{ij}

*λ*

_{3}

*τZ*

_{ij}

*λ*

_{4}

*τB*

_{ij}

*h*

_{i}, if one neglects the left side and use is made of (6c) for

^{θ}

_{i}

*h*

_{i}at level 3:where

*θ*

^{2}

*h*

_{i}at levels 2.5–2:whereand whereIn the following subsections, we will present a hierarchy of turbulence models for the PBL.

### b. Level-3 model

Since the level-2.5 and level-2 models catch the main features of the second-order closure models and are easy to use, they have become the most popular second-order closure models in the PBL community. We will concentrate on them in the sections below. Yet, the level-3 model has its own strength in that it produces countergradient heat fluxes, a phenomenon observed in the upper part of the convective PBL. In the appendix we will present the details of the level-3 model for completeness and for future reference.

### c. Level-2.5 model

*e*is solved from its prognostic equation (2f):The equation for the temperature variance

*θ*

^{2}

*u*

_{i}

*u*

_{j}

*u*

_{i}

*θ*

*G*

_{H}and

*G*

_{M}are defined asIn the above, the definitions of the stability functions

*S*

_{M}and

*S*

_{H}as well as the dimensionless gradients

*G*

_{M}and

*G*

_{H}are different than the corresponding definitions in the MY model. The transformation between the notations is straightforward:

In section 8 we will show that the MY model is a special case of the present model, and that the coefficients *s*_{2} and *d*_{5} are both nonzero in the new model and both zero in the MY model. Since *s*_{2} appears in the expression for *S*_{M} via Eq. (17a) and *d*_{5} appears in the expressions for both *S*_{M} and *S*_{H} via Eq. (18c), a “structural symmetry” can be seen in the new model while not in the MY model. By “structural symmetry,” we mean that: every *G*_{H} factor has a *G*_{M} counterpart, and both *G*_{H} and *G*_{M} enter with the same power. In the MY model, the term *s*_{2}*G*_{M} in (17a) and the term *d*_{5}*G*^{2}_{M}

### d. Realizability conditions for level-2.5 model

Realizability requirements are common to second-order closure models. For the present 2.5-level model, the two variables *G*_{M} and *G*_{H} must be limited to certain domains outside of which the model may produce unphysical results since some underlying assumptions (e.g., that departure from isotropy be small) may no longer be valid.

*G*

_{H}may be negative (unstable), zero (neutral), or positive (stable). Assuming that production equals dissipation for the turbulence kinetic energy

*e*[see Eq. (22) below], and taking the limit

*G*

_{M}→ 0 and noticing that

*G*

_{M}is always nonnegative, we have

*S*

_{H}

*G*

_{H}

*G*

_{H}

*G*

_{H}is −10.8; the negative value indicates that it occurs in the unstable region.

*G*

_{M},Although Eq. (21b) can be solved exactly, one may use the following approximate expression based on the fact that the terms containing

*s*

_{2}and

*d*

_{5}are relatively small,

### e. Level-2 model

*e,*Eq. (13), reduces to

*S*

_{M}

*G*

_{M}

*G*

_{H}

*G*

_{M}

*S*

_{H}

*G*

_{M}

*G*

_{H}

*G*

_{H}

*G*

_{M}(or for

*G*

_{H}) that depends on only one parameter, the gradient Richardson number,Equation (22) then becomeswhere

It is important to check the consistency of (21c) with (23b). At level 2, the results are presented in Fig. 1, while the use of the MY model gives rise to the results presented in Fig. 2. It is apparent that in the present model, *G*_{M} is smaller than *G*^{max}_{M}_{c} and thus the model is realizable. On the other hand, the MY *G*_{M} is larger than *G*^{max}_{M}

Substituting the *G*_{M} solved from (23b) into (17a,b) we can further plot the stability functions *S*_{M} and *S*_{H} as functions of Ri (Figs. 3 and 4).

## 6. Critical Richardson number

_{c}, beyond which stable stratification effectively suppresses the turbulence, can be found by considering the limit

*e*→ 0, that is,

*G*

_{M}→ ∞. In this limit, Eq. (23b) is satisfied only if the coefficient of the quadratic term vanishes, which yieldsUsing the model constants determined in section 7, we obtain

_{c}

_{c}∼ 0.2, there is a variety of data that are in favor of a Ri

_{c}of order one. Early laboratory data by Taylor (as cited in Monin and Yaglom 1971) showed that turbulent exchange exists even when Ri > 1. Webster (1964) and Young (1975)'s laboratory measurements showed that mixing persists up to Ri ∼ 1. In the oceanic PBL, Martin (1985) showed that Ri ∼ 1 is needed to obtain the correct mixed layer depth at Papa and November stations. More recently, direct numerical simulation (DNS; Gerz et al. 1989) and LES (e.g., Wang et al. 1996; Kosovic and Curry 2000) show that turbulence exists up to Ri ∼ 1. Historically, the criterionwas established by Miles (1961) and Howard (1961) on the basis of linear stability analysis. However, when nonlinear interactions were included, Abarbanel et al. (1984) showed that the sufficient and necessary condition for stability is not (24c) butwhich is in agreement with our result (24b).

The numerical value of Ri_{c} given by (24b) is a consequence of the closure parameterizations and the values of the model constants via Eq. (24a). It is to be understood that different choices of the (often scattered) data underlying the model constants may lead to somewhat different values of Ri_{c}, and the best we can do is to choose the ones we believe are the best, guided by the theoretical, nonlinear analysis (Abarbanel et al. 1984) and numerous LES and laboratory results, which indicate that Ri_{c} should be of order unity (also see Strang and Fernando 2001). Therefore, the value 0.96 in (24b) should be regarded as a suggestion and is subject to some changes when more data become available.

## 7. Determination of model constants

*τ*

_{pυ},

*τ*

_{pθ}and

*τ*

_{θ};

*τ*

_{pυ}, and

*τ*

_{pθ}are the timescales that enter the first term in the pressure correlations (thus, the subscript

*p*) for the velocity and temperature fields. In the phenomenological models these are known as the Rotta terms and their ratios to the dynamical timescale

*τ*were considered adjustable parameters. In fact, many previous higher-order PBL models determine these timescale ratios empirically (for a summary, see Wichmann and Schaller 1986). In the present study we take a new approach: instead of treating these parameters as free, we employ the expressions from a recent theoretical turbulence model that was based in part on renormalization group (RNG) methods and whose predictions were tested on a variety of flows (Canuto and Dubovikov 1996a,b, 1997):where

*σ*

_{t0}is the turbulent Prandtl number in neutral flows and will be determined later on. Applying (25a) in (9b), (10c), and (12c) giveswhere the value of

*γ*

_{1}is also given by RNG.

*λ*

_{2},

*λ*

_{3}, and

*λ*

_{4}, we adopt the following expressions (Shih and Shabbir 1992; Canuto 1994):where the value of

*β*

_{5}is given by RNG. Substituting (25a)–(25c) in (9b) yields

*λ*

_{2}

*λ*

_{3}

*λ*

_{4}

*ϵ*(the dissipation rate of

*e*) aswhich corresponds towhere the dissipation length scale ℓ ∼

*κz*as

*z*∼ 0 and the constant

*B*

_{1}is defined as

*B*

_{1}=

*q*

^{3}/

*u*

^{3}

_{∗}

*u*∗ is the friction velocity, and the value of

*B*

_{1}must be determined. In the neutral surface layer (taking the mean wind direction as the

*x*direction), we derive from (15e) that

*B*

_{1}is related to the values of

*λ*

_{1},

*λ*

_{2}, and

*λ*

_{3}:where we have used the fact that

*G*

_{M}∼

*B*

^{4/3}

_{1}

*B*

_{1}is different from the commonly used value of 16.6, which is determined in Mellor and Yamada (1982) by averaging several different data quoted in their Table 1. When different and/or new data are used in the averaging process, a new value of

*B*

_{1}may be obtained. For example, Enger (1986) uses

*B*

_{1}= 27 derived from Kansas spectra (Kaimal et al. 1972) and other laboratory data. A value of 27.4 was obtained for

*B*

_{1}by Nieuwstadt (1985) and by Andrén and Moeng (1993), a value of 22.6 was used by Therry and Lacarrére (1983) and a value of 24 is used by Nakanishi (2001). Instead of trying to determine

*B*

_{1}directly from the scattered data, we look into how the value of

*B*

_{1}relates to the values of

*λ*

_{1},

*λ*

_{2}, and

*λ*

_{3}[Eq. (27)]. The value of

*λ*

_{1}is determined from the renormalization group theory presented in Canuto and Dubovikov (1996a,b; 1997). The values of

*λ*

_{2}and

*λ*

_{3}are from theoretical formulations that are shown to be consistent with measured data (Shih and Shabbir 1992; Canuto 1994). The value 19.3 for

*B*

_{1}may be considered a compromise between the MY value (16.6) and the subsequent larger values.

*λ*

_{6},

*λ*

_{7}, and

*σ*

_{t0}, we need some auxiliary relations. First, from (15g), (15i), an expression for the ratio of the vertical and longitudinal heat fluxes can be derived,where

*σ*

_{t}≡

*S*

_{M}/

*S*

_{H}is the turbulent Prandtl number. Webster's (1964) experimental data show that this ratio approaches unity as Ri ∼ 0,Second, in a near-neutral surface layer, from (15g), (15i), we obtainUsing (10c), (28b), and (28c), we obtain,and

*λ*

_{6}and

*λ*

_{7}can be obtained using (28d) in (10c). We still need to determine a value for

*σ*

_{t0}in a consistent manner. From the third expression of (25b) and (28b)–(28d),

*σ*

_{t0}is found to be related to

*B*

_{1},

*λ*

_{2}and

*λ*

_{3}as follows:So it follows that:To summarize, the basic model constants determined above are presented in Table 1.

The other useful constants which can be calculated using Table 1 and Eqs. (18d) and (23c) are listed in Table 2.

## 8. Comparison with Mellor–Yamada model and experimental data

### a. Mellor–Yamada model

*A*

_{1},

*B*

_{1},

*A*

_{2}and

*B*

_{2}are determined by Mellor and Yamada to be

*A*

_{1}

*B*

_{1}

*A*

_{2}

*B*

_{2}

_{c}

### b. Comparison with measured data in neutral PBL

*υ*

^{2}

*w*

^{2}

*w*

^{2}

*υ*

^{2}

*υ*

^{2}

*w*

^{2}

*υ*

^{2}

*w*

^{2}

*υ*

^{2}

*w*

^{2}

*w*

^{2}

*υ*

^{2}

*υ*

^{2}

*w*

^{2}

*λ*

_{2}=

*λ*

_{3}, which makes

*υ*

^{2}

*w*

^{2}

*λ*

_{2}and

*λ*

_{3}are two independent parameters, and we choose to determine them according to Shih and Shabbir (1992)'s expressions that are derived from theoretical considerations. In Table 3 we compare the result of the present model in the neutral PBL with the measured data used by Mellor and Yamada (1982) and by Nieuwstadt (1985).

Since the difference between *υ*^{2}*w*^{2}*q*^{2} according to (31a)–(31c), and since *q*^{2} typically decreases with height and nearly vanishes near the top of the PBL, *υ*^{2}*w*^{2}

The predicted value for *u*^{2}*u*^{−2}_{∗}*z*/*h* (*h* is the PBL height) increases from the surface, *u*^{2}*υ*^{2}*w*^{2}*u*^{2}*u*^{−2}_{∗}*z*/*h* < 0.1; Figs. 26 and 27 of Khurshudyan et al. (1981), in which while *u*^{2}*υ*^{2}*w*^{2}*z*/*h* < 0.1].

### c. Comparison with measured and LES data in stratified flows

The turbulent Prandtl number, *σ*_{t} = *K*_{M}/*K*_{H}, is one of the important parameters of turbulence. We compare the inverse of *σ*_{t} as a function of the gradient Richardson number Ri resulting from both the present model and the MY model with the experimental data of Webster (1964). It is clear that turbulence in the stably stratified flow exists well beyond the MY critical value Ri ≈ 0.2. According to the experimental data, the critical value of Ri should be of order unity, and the present model falls within the range of the measured data (Fig. 5).

We also compare the vertical and lateral heat flux ratio −*wθ**uθ*_{c} = 0.96, in agreement with the data (Fig. 6).

*u*∗ and

*wθ*

_{s}are the friction velocity and the surface potential temperature flux, respectively, and

*S*is the shear given by Eq. (18b). Businger et al. (1971) analyzed the Kansas data in the constant flux surface layer and expressed Φ

_{m}and Φ

_{h}as functions of the dimensionless height

*ζ,*which is the ratio between the height

*z*and the Monin–Obukhov length

*L,*whereIn deriving (33a–b) Businger et al. assumed

*κ*= 0.35, where

*κ*is the von Kármán's constant. Högström (1988) subsequently modified Businger et al.'s formula with the more commonly accepted values for von Kármán's constant

*κ*= 0.4 and for Φ

_{h}at neutrality, (Φ

_{h})

_{ζ=0}= 0.95:The MY model (Mellor 1973; Mellor and Yamada 1982), by assuming ℓ =

*κz*in the surface layer, matches the original Businger et al.'s formula very well except for Φ

_{m}in the unstable region (

*ζ*< 0), where the MY model underestimates the Kansas data by about 50%. Recently Nakanishi (2001) has shown that in the surface layer, as indicated by the LES data, ℓ depends on

*ζ*as follows:In the present level-2 model, the expressions for Φ

_{m}and Φ

_{h}in terms of

*ζ*(via Ri) as well as ℓ/(

*κz*) are as follows:

Using Eqs. (35)–(36), we plot in Figs. 7 and 8 Φ_{m} and Φ_{h} versus *ζ* for both unstable and stable conditions, and compare them with the Kansas data as originally formulated by Businger et al. (1971) and as modified by Högström (1988). We also plot the results of the MY model using the length scale (35). The comparison shows that:

- In the unstable region (
*ζ*< 0), the present model (solid line) improves significantly the MY model (dotted line) for Φ_{m}, and improves Φ_{h}slightly. - In the stable region (
*ζ*> 0), both the present model and the MY model fall within the (scattered) data regions. For more stable cases, however, the two models will further diverge, as we will show below.

In recent years, several LESs have provided ^{−1}_{m}^{−1}_{h}^{−1}_{m}^{−1}_{h}

In our simulation, we use the level-2 model since we are particularly interested in the behavior of the model when the gradient Richardson number Ri varies. While the level-2.5 and -3 models depend on two independent parameters, *G*_{M} and *G*_{H}, the level-2 model depends on only one parameter, Ri. In the PBL we chose to simulate, the diffusion terms are very small (see Fig. 11 of Kosovic and Curry 2000). We have also run the level-2.5 and -3 models using the usual downgradient approximations for the diffusion terms, and the results are very close to those from the level-2 model.

In fact, to see the full benefits of level-2.5 and -3 models, one needs to parameterize the third moments much better than by the downgradient approximation. As stated in the introduction, in the present study we concentrate on the improvements due to the new pressure correlation parameterizations, and leave the third moment parameterizations for future study. Thus the level-2 model with a commonly used length scale formula, Eq. (5e), is most appropriate for testing the model.

In Figs. 9 and 10 we plot ^{−1}_{m}^{−1}_{h}

The differences between the present model and the LES results are probably due to the neglect of the diffusion terms and the imperfect parameterization of the turbulence length scale, and search for better parameterizations of these two crucial components of the closure modeling should be among the subjects of future studies.

The PBL height is one of the most important quantities in any PBL modeling. The PBL height is usually defined as the height at which the turbulent kinetic energy or the magnitude of the momentum flux decreases to a small fraction of the corresponding surface value; or it may be defined as the height at which the (positive) temperature gradient reaches a certain value from below. In any case, the top of the PBL lies in a region where the turbulence is stably stratified and, given the mean profiles of the wind and the temperature (and thus given Ri), a higher intensity level of turbulence yields a greater PBL height. The MY model, however, underestimates the PBL height (Yamada and Mellor 1975). Since the present model predicts larger critical Richardson number and produces more turbulence for a given Richardson number, greater PBL heights can be achieved (Fig. 11).

## 9. Conclusions

With the application of the updated expressions for the pressure–velocity and pressure–temperature correlations and the use of the turbulence timescale ratios fixed by recent RNG, we have derived a second-order closure turbulence model to describe the PBL.

One of the improvements brought about by the present model is that it distinguishes the vertical and the lateral components of the turbulence kinetic energy in neutral PBLs without the complexity of the wall functions, something that was not achieved by previous second-order closure PBL models.

A main feature of the new model is that it yields a critical Richardson number (Ri_{c}) of order unity, rather than ∼0.2, as given by most previous models. The larger critical Richardson number is in agreement with measured and LES data and the stability analysis that includes nonlinear interactions. The new model compares better than the previous models with the Kansas data as analyzed by Businger et al. (1971) and modified by Högström (1988) for both the unstable case (Ri < 0) and the stable case when Ri < 0.2. While most previous models predict no turbulence for Ri > 0.2, the present model reproduces closely the LES and laboratory data for Richardson numbers up to order unity.

In addition, the new model produces greater PBL height than the previous models.

The authors thank Drs. B. Kosovic and J.A. Curry for providing their LES data to be used in this paper.

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# APPENDIX

## The Level-3 PBL Model

*θ*

^{2}

*wθ*

*D*is of the same form as in (18c). The structure of the stability function

*S*

_{M}differs from the

*S*

_{M}in the level-2.5 model (17a) by an extra term,where

*λ*

_{0}is a new model constant in the level-3 model, and

*s*

_{3}is a new derived constant. Note that in (A3) we use

*S*

^{′}

_{H}

*S*

_{H}for the stability function because of the existence of the countergradient term

*γ*

_{c}. The form of the function

*S*

^{′}

_{H}

*S*

_{H}in the level-2.5 model (17b). The model constants

*B*

_{1}and

*λ*′

*s*are the same as in Table 1 except that now

*λ*

_{8}= 0. The expressions for the derived constants

*d*′

*s*and

*s*′

*s*are the same as in (18d) with

*λ*

_{8}= 0.

Basic model constants

Derived constants. Useful constants calculated using Table 1 and Eqs. (18d) and (23c)

Measured data and present model prediction in neutral PBL