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  • View in gallery

    Solid line: GM as a function of the gradient Richardson number Ri, obtained from the present model at level 2, Eq. (23b); dashed line: GmaxM as a function of Ri, obtained from Eq. (21c)

  • View in gallery

    Similar to Fig. 1 but for the MY model

  • View in gallery

    The stability function SM vs the gradient Richardson number Ri. The solid line represents the present model; the dotted line, the MY model. Note that the definitions of SM and SH and those in the MY model differ by a constant [see Eq. (19a)]

  • View in gallery

    Same as Fig. 3 but for the stability function SH

  • View in gallery

    The inverse turbulent Prandtl number σ−1t (normalized by its value for neutral stratification) vs the gradient Richardson number. The solid line is the result of the present model at level 2. The dotted line represents the level-2 MY model. The experimental data by Webster (1964) are redrawn here as filled circles. The present model yields a much larger critical Richardson number (≈1) than the Mellor–Yamada model (≈0.2)

  • View in gallery

    Ratio of the rates of heat transport in the w direction (vertical) and the u direction (horizontal, along the mean flow), −/, vs the Richardson number. The solid line represents the result of the present model, while the dotted line represents the MY model. The experimental data (Webster 1964) are redrawn here as filled circles

  • View in gallery

    The nondimensional shear Φm as a function of ζ = z/L. The solid line represents the results using the present model, while the dotted line corresponds to the MY model. The squares represent the Kansas data formulated by Businger et al. (1971), while the triangles represent Businger et al.'s formula modified by Högström (1988)

  • View in gallery

    Same as Fig. 7 but for the nondimensional potential temperature gradient Φh

  • View in gallery

    The reciprocal of the nondimensional shear, Φ−1m, as a function of the gradient Richardson number. The crosses represent the LES simulation of Kosovic and Curry (2000), case NLHRB, at hour 12. The solid line represents simulation results using the present model, while the dotted line, simulation results using the MY model. The triangles represent the Kansas data formulated by Businger et al. (1971) and modified by Högström (1988)

  • View in gallery

    Similar to Fig. 9 but for the reciprocal of the nondimensional potential temperature gradient, Φ−1h

  • View in gallery

    PBL height as a function of the dimensionless time tfc, where fc is the Coriolis parameter. Cross: LES result; solid line: present model result; dotted line: MY model result

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An Improved Model for the Turbulent PBL

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  • 1 NASA Goddard Institute for Space Studies, New York, New York
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Abstract

Second-order turbulence models of the Mellor and Yamada type have been widely used to simulate the planetary boundary layer (PBL). It is, however, known that these models have several deficiencies. For example, assuming the production of the turbulent kinetic energy equals its dissipation, they all predict a critical Richardson number that is about four times smaller than the large eddy simulation (LES) data in stably stratified flows and are unable to distinguish the vertical and lateral components of the turbulent kinetic energy in neutral PBLs, and they predict a boundary layer height lower than expected.

In the present model, three new ingredients are employed: 1) an updated expression for the pressure–velocity correlation, 2) an updated expression for the pressure–temperature correlation, and 3) recent renormalization group (RNG) expressions for the different turbulence timescales, which yield

1) a critical Richardson number of order unity in the stably stratified PBL (at level 2 of the model),

2) different vertical and lateral components of the turbulent kinetic energy in the neutral PBL obtained without the use of the wall functions,

3) a greater PBL height,

4) closer comparisons with the Kansas data in the context of the Monin–Obukhov PBL similarity theory, in both stable and unstable PBLs, and

5) more realistic comparisons with the LES and laboratory data.

Additional affiliation: Department of Applied Physics and Mathematics, Columbia University, New York, New York

Corresponding author address: Dr. V. M. Canuto, NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025. Email: acvmc@giss.nasa.gov.

Abstract

Second-order turbulence models of the Mellor and Yamada type have been widely used to simulate the planetary boundary layer (PBL). It is, however, known that these models have several deficiencies. For example, assuming the production of the turbulent kinetic energy equals its dissipation, they all predict a critical Richardson number that is about four times smaller than the large eddy simulation (LES) data in stably stratified flows and are unable to distinguish the vertical and lateral components of the turbulent kinetic energy in neutral PBLs, and they predict a boundary layer height lower than expected.

In the present model, three new ingredients are employed: 1) an updated expression for the pressure–velocity correlation, 2) an updated expression for the pressure–temperature correlation, and 3) recent renormalization group (RNG) expressions for the different turbulence timescales, which yield

1) a critical Richardson number of order unity in the stably stratified PBL (at level 2 of the model),

2) different vertical and lateral components of the turbulent kinetic energy in the neutral PBL obtained without the use of the wall functions,

3) a greater PBL height,

4) closer comparisons with the Kansas data in the context of the Monin–Obukhov PBL similarity theory, in both stable and unstable PBLs, and

5) more realistic comparisons with the LES and laboratory data.

Additional affiliation: Department of Applied Physics and Mathematics, Columbia University, New York, New York

Corresponding author address: Dr. V. M. Canuto, NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025. Email: acvmc@giss.nasa.gov.

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 (Ric) 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, w2 and υ2; in fact, they yield identical expressions for the two, while experiments consistently show that the vertical component is much smaller than the lateral one (Table 1 of Mellor and Yamada 1982; Nieuwstadt 1985).

As we will show below, these deficiencies are associated with the oversimplification of the parameterizations of the pressure–velocity correlation Πij and pressure–temperature correlation Πθi, which will be corrected by adopting more complete expressions. Both Πij and Πθi have been shown to contain a slow (return-to-isotropy) part and a rapid part (Launder et al. 1975; Lumley 1978). The rapid parts of both Πij and Πθi contain velocity terms related to the mean strain-rate tensor Sij, and the vorticity tensor Rij, as well as buoyancy terms related to the heat fluxes. In addition, the rapid part of Πθi also contains a term related to the temperature variance θ2. By contrast, MY models of Πij include only the slow part and some of the rapid part (the term proportional to eSij, where e is the turbulent kinetic energy); for Πθi, only the slow part is included. Since each of these missing terms represents a specific physical process, it seems appropriate and necessary to incorporate them in the model formulation, as we do in the present paper.

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:

1) mean velocity, Ui:
i1520-0469-59-9-1550-e1a
2) mean potential temperature, Θ:
i1520-0469-59-9-1550-e1b
where
i1520-0469-59-9-1550-e1c
Here ui is the ith component of the turbulent velocity fluctuation, gi = (0, 0, g) is the gravitational acceleration, P is the mean pressure, ρ is the mean density, Ωj is the rotation of the earth, τij is the Reynolds stress, and hi is the heat flux.
In the PBL, several approximations can be made to the equations for the mean wind and temperature. In Eq. (1a), the horizontal pressure gradient can be expressed in terms of the mean geostrophic wind components Ug and Vg as follows:
i1520-0469-59-9-1550-e1d
and the rotation term can be approximated as
ϵijkjUkfcϵij3Uj
where x, y, and z are the eastward, northward, and vertical directions, respectively, fc = 2Ω sinϕ 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,
i1520-0469-59-9-1550-e1f
where α is the volume expansion coefficient.
The equations for the eastward and northward horizontal mean wind components U and V and for the mean potential temperature Θ in the PBL can then be written as
i1520-0469-59-9-1550-e1g

3. Turbulence equations

1) Reynolds stresses, τij:
i1520-0469-59-9-1550-e2a
where
i1520-0469-59-9-1550-e2b

Here, Πij is the pressure–velocity correlation tensor, ν is the molecular viscosity, ϵ is the dissipation rate of the turbulent kinetic energy e, and Dij is the diffusion term.

2) Turbulent kinetic energy e:
i1520-0469-59-9-1550-e2e
3) Heat flux, hi:
i1520-0469-59-9-1550-e3a
where
i1520-0469-59-9-1550-e3b
where Πθi is the pressure–temperature correlation, and Dhi is the diffusion of the heat flux hi.
4) Temperature variance, θ2:
i1520-0469-59-9-1550-e4a
where
i1520-0469-59-9-1550-e4b
where χ is the molecular conductivity, Dθ is the diffusion of the temperature variance, and ϵθ 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 ϵij and ϵθ. 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 Πij and Πθi, which will be shown to improve the PBL model results.

4. Turbulence closure

a. ϵ and ϵθ

Equations (2f) and (4a) contain the two turbulence variables,
ϵ,ϵθ
which represent the rates of dissipation of e and θ2, and are contributed mostly by small scales with small energy content but large vorticity. On the other hand, e and θ2 are contributed mostly by the large scales with most of the energy and little vorticity.
Even though exact dynamic equations for ϵ and ϵθ 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)
i1520-0469-59-9-1550-e5b
where Λ is the dissipation length scale. Rewriting (5b) in terms of Λ and τ defined as
i1520-0469-59-9-1550-e5c
one has
i1520-0469-59-9-1550-e5d
In 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 whereby
i1520-0469-59-9-1550-e5e
where B1 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).
The differential equation for ϵθ is even more difficult to calibrate than the ϵ equation and also too complicated to use. We use instead the parameterization
i1520-0469-59-9-1550-e5f
and we will determine the timescale τθ in section 7.

b. e and θ2

The dynamic equations for e and θ2 are given by Eqs. (2f) and (4a). In the so-called level-3 model these two differential equations are solved. In the level-2.5 model, the θ2 equation is reduced to an algebraic equation by neglecting the storage, advection, and diffusion terms. In the level-2 model one similarly simplifies the e equation by assuming that production of e equals its dissipation:
PbPsϵ,
where Ps,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 dependence
i1520-0469-59-9-1550-e5h
and thus the solution of (5g) does not yield e or ϵ separately but only their ratio τ,
τNf
The function 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

The pressure correlation terms Πij and Πθi in Eqs. (2b) and (3b) contain three distinct contributions due to 1) turbulence self-interactions (the return-to-isotropy or slow part), 2) mean shear–turbulence interactions (a rapid part), and 3) buoyancy–turbulence interactions (also a rapid part). The most complete models for Πij and Πθi are given by (Launder et al. 1975; Zeman and Lumley 1979):
i1520-0469-59-9-1550-e6a
where
i1520-0469-59-9-1550-e6b
where bij is the traceless Reynolds stress tensor defined as follows:
i1520-0469-59-9-1550-e6d
The other tensors are defined as follows:
i1520-0469-59-9-1550-e6e
where Sij and Rij are shear and vorticity, respectively.
In most past second-order turbulence models for the PBL, the pressure correlations terms were parameterized much less completely than in (6a)–(6c). For Πij, the MY (1982; Mellor 1973) and Kantha and Clayson (1994) models only consider:
i1520-0469-59-9-1550-e7a
for Πθi, the MY model contains only the terms
θ(1)iτ−1hiθ(2)iθ(3)i
Namely, the MY model includes only the slow terms and one single rapid term (the first term in the expression of Π(2)ij); most of the rapid terms are neglected, and no buoyancy effects are included. The models by Kantha and Clayson (1994) and D'Alessio et al. (1998) improve the parameterization for Πθi, but the Πθ(2)i term is still missing,
θ(1)iτ−1hiθ(2)iθ(3)iγ1βiθ2
More complete parameterizations of both Πij and Πθi were used by Gambo (1978), Yamada (1985), and Nakanishi (2001), but α1 and α2 were either taken to be identical or set to zero in Eq. (6b), and Sij and Rij were set to have the same coefficient in Eq. (6c).

In section 5, we will derive new expressions for the stability functions (SM and SH) 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

Combining (2a) and (2f) and using (5c), we obtain the equation for bij, defined in Eq. (6d):
i1520-0469-59-9-1550-e8a
where
i1520-0469-59-9-1550-e8b
Assuming that the left side of (8a) can be neglected and employing (6b) for the pressure–velocity correlation Πij, one obtains the following algebraic equation for bij
bijλ1eτSijλ2τijλ3τZijλ4τBij
where
i1520-0469-59-9-1550-e9b
These model constants will be given in section 7. Similarly, in the prognostic equation (3a) for the heat flux hi, if one neglects the left side and use is made of (6c) for Πθi, one obtains the algebraic equation for hi at level 3:
i1520-0469-59-9-1550-e10a
where
i1520-0469-59-9-1550-e10b
At levels 2.5 and 2, we further simplify the problem by neglecting the left side in the prognostic equation for θ2, Eq. (4a), to obtain the algebraic equation
i1520-0469-59-9-1550-e11
Substituting (11) into (10a), we obtain the algebraic equation for hi at levels 2.5–2:
i1520-0469-59-9-1550-e12a
where
i1520-0469-59-9-1550-e12b
and where
i1520-0469-59-9-1550-e12c
In 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

In the level-2.5 model, the turbulent kinetic energy e is solved from its prognostic equation (2f):
i1520-0469-59-9-1550-e13
The equation for the temperature variance θ2 is
i1520-0469-59-9-1550-e14
From the algebraic equations for uiuj and uiθ, Eqs. (9a) and (12a), we obtain:
i1520-0469-59-9-1550-e15a
Equations (15a)–(15i) can be solved using symbolic algebra. The results are
i1520-0469-59-9-1550-e16a
where GH and GM are defined as
i1520-0469-59-9-1550-e18a
In the above, the definitions of the stability functions SM and SH as well as the dimensionless gradients GM and GH are different than the corresponding definitions in the MY model. The transformation between the notations is straightforward:
i1520-0469-59-9-1550-e19a

In section 8 we will show that the MY model is a special case of the present model, and that the coefficients s2 and d5 are both nonzero in the new model and both zero in the MY model. Since s2 appears in the expression for SM via Eq. (17a) and d5 appears in the expressions for both SM and SH 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 GH factor has a GM counterpart, and both GH and GM enter with the same power. In the MY model, the term s2GM in (17a) and the term d5G2M in (18c) are missing.

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 GM and GH 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.

Let us first consider the limitation on buoyancy. Here GH 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 GM → 0 and noticing that GM is always nonnegative, we have
SHGHGH
Substituting Eq. (17b) into Eq. (20a) yields the relation
i1520-0469-59-9-1550-e20b
For the model constants used here (see section 7), this minimum value of GH is −10.8; the negative value indicates that it occurs in the unstable region.
Next, we examine the limitation on the shear number. Following Hassid and Galperin (1983), who argue that an increase of shear should not result in a decrease of the normalized momentum flux, we apply the following condition,
i1520-0469-59-9-1550-e21a
Using Eqs. (16)–(18), Eq. (21a) can be reduced to a cubic inequality in GM,
i1520-0469-59-9-1550-e21b
Although Eq. (21b) can be solved exactly, one may use the following approximate expression based on the fact that the terms containing s2 and d5 are relatively small,
i1520-0469-59-9-1550-e21c

e. Level-2 model

If we assume that production equals dissipation, the differential equation for e, Eq. (13), reduces to
SMGMGHGMSHGMGHGH
which can be rewritten as an equation for GM (or for GH) that depends on only one parameter, the gradient Richardson number,
i1520-0469-59-9-1550-e23a
Equation (22) then becomes
i1520-0469-59-9-1550-e23b
where
i1520-0469-59-9-1550-e23c

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, GM is smaller than GmaxM for all Ri < Ric and thus the model is realizable. On the other hand, the MY GM is larger than GmaxM for Ri ≥ 0.064, indicating that the MY model at level 2 is not compatible with Hassid and Galperin (1983)'s condition even for moderate Richardson numbers.

Substituting the GM solved from (23b) into (17a,b) we can further plot the stability functions SM and SH as functions of Ri (Figs. 3 and 4).

6. Critical Richardson number

In the level-2 model, the critical Richardson number Ric, beyond which stable stratification effectively suppresses the turbulence, can be found by considering the limit e → 0, that is, GM → ∞. In this limit, Eq. (23b) is satisfied only if the coefficient of the quadratic term vanishes, which yields
i1520-0469-59-9-1550-e24a
Using the model constants determined in section 7, we obtain
c
Although most previous second-order closure models give Ric ∼ 0.2, there is a variety of data that are in favor of a Ric 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 criterion
i1520-0469-59-9-1550-e24c
was 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) but
which is in agreement with our result (24b).

The numerical value of Ric 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 Ric, 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 Ric 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

A critical part in the determination of the closure parameters (defined in 9b, 10c, and 12c) is the turbulence timescale ratios τ, τ and τθ; τ, and τ 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):
i1520-0469-59-9-1550-e25a
where σt0 is the turbulent Prandtl number in neutral flows and will be determined later on. Applying (25a) in (9b), (10c), and (12c) gives
i1520-0469-59-9-1550-e25b
where the value of γ1 is also given by RNG.
To determine λ2, λ3, and λ4, we adopt the following expressions (Shih and Shabbir 1992; Canuto 1994):
i1520-0469-59-9-1550-e25c
where the value of β5 is given by RNG. Substituting (25a)–(25c) in (9b) yields
λ2λ3λ4
We parameterize ϵ (the dissipation rate of e) as
i1520-0469-59-9-1550-e26a
which corresponds to
i1520-0469-59-9-1550-e26b
where the dissipation length scale ℓ ∼ κz as z ∼ 0 and the constant B1 is defined as B1 = q3/u3, where u∗ is the friction velocity, and the value of B1 must be determined. In the neutral surface layer (taking the mean wind direction as the x direction), we derive from (15e) that B1 is related to the values of λ1, λ2, and λ3:
i1520-0469-59-9-1550-e27
where we have used the fact that GMB4/31. This value of 19.3 for B1 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 B1 may be obtained. For example, Enger (1986) uses B1 = 27 derived from Kansas spectra (Kaimal et al. 1972) and other laboratory data. A value of 27.4 was obtained for B1 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 B1 directly from the scattered data, we look into how the value of B1 relates to the values of λ1, λ2, and λ3 [Eq. (27)]. The value of λ1 is determined from the renormalization group theory presented in Canuto and Dubovikov (1996a,b; 1997). The values of λ2 and λ3 are from theoretical formulations that are shown to be consistent with measured data (Shih and Shabbir 1992; Canuto 1994). The value 19.3 for B1 may be considered a compromise between the MY value (16.6) and the subsequent larger values.
To determine the values of λ6, λ7, and σt0, we need some auxiliary relations. First, from (15g), (15i), an expression for the ratio of the vertical and longitudinal heat fluxes can be derived,
i1520-0469-59-9-1550-e28a
where σtSM/SH is the turbulent Prandtl number. Webster's (1964) experimental data show that this ratio approaches unity as Ri ∼ 0,
i1520-0469-59-9-1550-e28b
Second, in a near-neutral surface layer, from (15g), (15i), we obtain
i1520-0469-59-9-1550-e28c
Using (10c), (28b), and (28c), we obtain,
i1520-0469-59-9-1550-e28d
and λ6 and λ7 can be obtained using (28d) in (10c). We still need to determine a value for σt0 in a consistent manner. From the third expression of (25b) and (28b)–(28d), σt0 is found to be related to B1, λ2 and λ3 as follows:
i1520-0469-59-9-1550-e29a
So it follows that:
i1520-0469-59-9-1550-e29b
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

The MY model (Mellor and Yamada 1982) corresponds to
i1520-0469-59-9-1550-e30a
Thus
i1520-0469-59-9-1550-e30b
where the constants A1, B1, A2 and B2 are determined by Mellor and Yamada to be
A1B1A2B2
which correspond to a set of value for the model constants in the present model
i1520-0469-59-9-1550-e30d
Substituting (30d) into (18d), (23c), and (24a) yields
c

b. Comparison with measured data in neutral PBL

One of the deficiencies of the MY model, as Mellor and Yamada pointed out themselves, is that in a neutral boundary layer, the model cannot distinguish υ2 and w2, the lateral and vertical components of the velocity variance, while experimental data consistently show that w2 is always significantly smaller than υ2. Shir (1973) and Gibson and Launder (1978) added additional terms to the pressure correlations to parameterize the wall effects, assuming that in proximity to the wall, the transfer of turbulence energy from the horizontal to the vertical components is altered as the vertical extent of the eddies is suppressed. In this way υ2 and w2 can accordingly be differentiated. The present model, however, offers an alternative that will be able to, at least partially, account for the difference between υ2 and w2, without resorting to adding wall terms to the pressure correlations. In fact, the measured data indicate that the inequality of υ2 and w2 may be mainly not due to the wall effects, as shown by Fig. 6 of Grant (1992), in which the ratio w2/υ2 is roughly around 0.5, even far away from the boundaries; the ratio is never 1 at any height within the PBL. The present model employs the standard, advanced pressure correlations (without wall terms) into the model closure, which naturally allow υ2 and w2 to be different. To see this we assume production equals dissipation in a neutral boundary layer, reducing (15a)–(15c) of the present model to
i1520-0469-59-9-1550-e31a
In the MY model (and in all the second-closure PBL models known to us) λ2 = λ3, which makes υ2 = w2, while in the present model λ2 and λ3 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 υ2 and w2 is proportional to q2 according to (31a)–(31c), and since q2 typically decreases with height and nearly vanishes near the top of the PBL, υ2w2 also decreases and vanishes as height increases. Thus both the surface and the free flow cases are approximated.

The predicted value for u2u−2 is smaller than the quoted data for the following reason: the quoted data were taken mostly in the lower part of the surface layer, while (31a)–(31c) give some weight to data in the middle and upper parts of the surface layer. The profiles of the measured data show that when the scaled height z/h (h is the PBL height) increases from the surface, u2 decreases faster than υ2 and w2 [see, e.g., Fig. 1a of Andrén (1991), in which u2u−2 drops to below 3 at z/h < 0.1; Figs. 26 and 27 of Khurshudyan et al. (1981), in which while u2 decreases with height, υ2 and w2 actually increase slightly for z/h < 0.1].

c. Comparison with measured and LES data in stratified flows

The turbulent Prandtl number, σt = KM/KH, is one of the important parameters of turbulence. We compare the inverse of σt 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 −/ (as a function of Ri resulting from both the present model and the MY model with the experimental data of Webster (1964). Webster described the ratio as “(being) seen to fall catastrophically from unity in neutral conditions to only about 0.5 at Ri equal to 0.2 and even less for higher Richardson numbers.” The present model gives the critical Richardson number Ric = 0.96, in agreement with the data (Fig. 6).

It is also informative to examine the nondimensional shear and potential temperature gradients defined as
i1520-0469-59-9-1550-e32
where u∗ and 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 Φm and Φh as functions of the dimensionless height ζ, which is the ratio between the height z and the Monin–Obukhov length L,
i1520-0469-59-9-1550-e33a
where
i1520-0469-59-9-1550-e33c
In 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 Φh at neutrality, (Φh)ζ=0 = 0.95:
i1520-0469-59-9-1550-e34a
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 Φm 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:
i1520-0469-59-9-1550-e35
In the present level-2 model, the expressions for Φm and Φh in terms of ζ (via Ri) as well as ℓ/(κz) are as follows:
i1520-0469-59-9-1550-e36

Using Eqs. (35)–(36), we plot in Figs. 7 and 8 Φm and Φh 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:

  1. In the unstable region (ζ < 0), the present model (solid line) improves significantly the MY model (dotted line) for Φm, and improves Φh slightly.
  2. 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 Φ−1m and Φ−1h as functions of the gradient Richardson number Ri (e.g., Mason 1994; Brown et al. 1994; Andrén 1995; Kosovic and Curry 2000). The Wangara data have also been analyzed and the resulting Φ−1m and Φ−1h plotted (Carson and Richards 1978). In all these studies, turbulence exists with significant intensity around the commonly accepted value of the critical Richardson number 0.2, and extends up to Ri of order unity. Since the most recent LES by Kosovic and Curry (2000) use a more advanced subgrid model, we choose to compare with their results. We employ both the present model and the MY model to simulate the same stably stratified PBL used by Kosovic and Curry (2000), and compare the models results with their LES of the high-resolution case NLHRB, at hour 12, when a quasi-steady state is reached.

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, GM and GH, 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 Φ−1m and Φ−1h as functions of Ri. The graphs indicate that, in the context of Monin–Obukhov similarity theory, for Ri < 0.2, the present model recovers the observed Kansas data as analyzed by Businger et al. (1971) and modified by Högström (1988). For Ri > 0.2, the present model still produces significant turbulence, in agreement with the LES data by Kosovic and Curry (2000), which is consistent with the LES of Mason (1994), Brown et al. (1994), and Andrén (1995) and the Wangara data analyzed by Carson and Richards (1978). In the figures we also plot the results of the MY model, which fail to reproduce the turbulence beyond Ri = 0.2 found in the LES.

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 (Ric) 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.

Acknowledgments

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

In the level-3 PBL model, the turbulent temperature variance θ2 is solved from its prognostic equation [which replaces the algebraic equation (14)]:
i1520-0469-59-9-1550-ea1
From (10a)–(10b), the algebraic equation for the heat flux is
i1520-0469-59-9-1550-ea2
All the other algebraic equations for the Reynolds stress and the heat flux, except (15i), which is replaced by (A2), are the same as the level-2.5 model (15a)–(15h). We solve (15a)–(15h) and (A2) using symbolic algebra and the results are
i1520-0469-59-9-1550-ea3
where
i1520-0469-59-9-1550-ea4
is the countergradient term, which is absent in the level-2.5 model and D is of the same form as in (18c). The structure of the stability function SM differs from the SM in the level-2.5 model (17a) by an extra term,
i1520-0469-59-9-1550-ea5
where
i1520-0469-59-9-1550-ea6
λ0 is a new model constant in the level-3 model, and s3 is a new derived constant. Note that in (A3) we use SH instead of SH for the stability function because of the existence of the countergradient term γc. The form of the function SH is the same as SH in the level-2.5 model (17b). The model constants B1 and λs are the same as in Table 1 except that now λ8 = 0. The expressions for the derived constants ds and ss are the same as in (18d) with λ8 = 0.
Fig. 1.
Fig. 1.

Solid line: GM as a function of the gradient Richardson number Ri, obtained from the present model at level 2, Eq. (23b); dashed line: GmaxM as a function of Ri, obtained from Eq. (21c)

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 2.
Fig. 2.

Similar to Fig. 1 but for the MY model

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 3.
Fig. 3.

The stability function SM vs the gradient Richardson number Ri. The solid line represents the present model; the dotted line, the MY model. Note that the definitions of SM and SH and those in the MY model differ by a constant [see Eq. (19a)]

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 4.
Fig. 4.

Same as Fig. 3 but for the stability function SH

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 5.
Fig. 5.

The inverse turbulent Prandtl number σ−1t (normalized by its value for neutral stratification) vs the gradient Richardson number. The solid line is the result of the present model at level 2. The dotted line represents the level-2 MY model. The experimental data by Webster (1964) are redrawn here as filled circles. The present model yields a much larger critical Richardson number (≈1) than the Mellor–Yamada model (≈0.2)

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 6.
Fig. 6.

Ratio of the rates of heat transport in the w direction (vertical) and the u direction (horizontal, along the mean flow), −/, vs the Richardson number. The solid line represents the result of the present model, while the dotted line represents the MY model. The experimental data (Webster 1964) are redrawn here as filled circles

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 7.
Fig. 7.

The nondimensional shear Φm as a function of ζ = z/L. The solid line represents the results using the present model, while the dotted line corresponds to the MY model. The squares represent the Kansas data formulated by Businger et al. (1971), while the triangles represent Businger et al.'s formula modified by Högström (1988)

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 8.
Fig. 8.

Same as Fig. 7 but for the nondimensional potential temperature gradient Φh

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 9.
Fig. 9.

The reciprocal of the nondimensional shear, Φ−1m, as a function of the gradient Richardson number. The crosses represent the LES simulation of Kosovic and Curry (2000), case NLHRB, at hour 12. The solid line represents simulation results using the present model, while the dotted line, simulation results using the MY model. The triangles represent the Kansas data formulated by Businger et al. (1971) and modified by Högström (1988)

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 10.
Fig. 10.

Similar to Fig. 9 but for the reciprocal of the nondimensional potential temperature gradient, Φ−1h

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Fig. 11.
Fig. 11.

PBL height as a function of the dimensionless time tfc, where fc is the Coriolis parameter. Cross: LES result; solid line: present model result; dotted line: MY model result

Citation: Journal of the Atmospheric Sciences 59, 9; 10.1175/1520-0469(2002)059<1550:AIMFTT>2.0.CO;2

Table 1.

Basic model constants

Table 1.
Table 2.

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

Table 2.
Table 3.

Measured data and present model prediction in neutral PBL

Table 3.
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