a. The third-order moments

A traditional closure approximation for the third moments in the foregoing equations involves representing these quantities as being entirely due to down-gradient diffusion (e.g., Stull 1988, 204). However, as noted, for example, by Zeman and Lumley (1976) and later Moeng and Wyngaard (1989), such a down-gradient diffusion assumption for the third moments, although adequate for stable and neutral conditions (when they are often relatively small), is inadequate for the convective regime in the planetary boundary layer (PBL). In the convective PBL, the transport of the third-order moments contributes significantly to the rate equations of the second moments. Therefore, an incorrect representation of the third-order moments will significantly degrade the accuracy of the fluxes and consequently of the mean state throughout the ABL, including the entrainment zone at its the top.

As an alternative approach we appeal to a convective mass-flux concept as a guide to representing the vertical transports of fluxes and variances of scalar quantities. As is demonstrated below, this leads to a parameterization of second-order moments in terms of nonlocal diffusivities and includes a counter-gradient term for the heat flux.

1) Parameterizing W ′²θ′ and W ′θ′²

The convective mass-flux concept has been used for parameterization of fluxes and variances in convectively active boundary layers in a number of studies (e.g., Betts 1973,1983; Wang and Albrecht 1990; Randal et al. 1992). It has also been employed in observational studies (e.g., Lenschow and Stephens 1982; Khasla and Greenhut 1985; Young 1988) and for analyses of the results of large eddy simulations (e.g., Moeng and Schumann 1991). Here we use the mass-flux concept as a guide to representing the third-moment terms w′²θ′ and w′θ′², which appear in Eqs. (8) and (10).

Consider an idealized convective circulation composed of rising and sinking branches, updrafts, and downdrafts. If c is a generic variable representing any scalar, then the area mean c̄ is given by

c̄ac_uac_d(11)

where a is the fractional area occupied by the updrafts; and c_u and c_d are the mean values of c within the updraft and the downdraft, respectively. Then, assuming “top hat” profiles, the turbulent fluxes of c due to the convective circulations can be written as

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Similarly, the transport of the turbulent flux of c can be written as

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Similarly, as demonstrated by Randal et al. (1992), the following expressions are obtained for the the variance of c and its transport,

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The skewness S_w of the vertical velocity can be written as

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Combining Eqs. (12)–(15) we obtain the following expressions for the third-order moments,

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Hence replacing c with θ in (17) and (18) we obtain

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In order to determine the usefulness of these expressions in convectively active conditions, we assume for simplicity that the skewness is constant and consider the forms of f_w and f_θ near the top of the surface layer (which we assume to be at z = 0.1 h where h is the depth of the boundary layer). Surface layer similarity theory (see Stull 1988, 370) and observations (Lenschow 1980) suggests that

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where c₁ = 1.3S_w, c₂ = 2.9S_w, and w∗ is the convective velocity scale given by

wβgw′θ′_sh^1/3(23)

where w′θ′_s is the kinematic surface heat flux and θ∗ = w′θ′_s;t2/w∗ is the convective temperature scale. For neutral and stable conditions both w∗ and θ∗ are set to zero.

In Eq. (21) we retain the (z/h)^1/3 dependence of w′² in the similarity relation instead of evaluating it at the top of the surface layer, in order to be consistent with results of LES simulations (Moeng and Wyngaard 1989) for w′²θ′. However, these simulations also suggest that θ′² is nearly independent of height in the PBL, which is consistent with Eq. (22). In fact, using (21) and (22) in Eqs. (19) and (20), respectively, makes the expressions for w′²θ′ and w′θ′² consistent with LES results if we choose c₁ = 0.9 and c₂ = 2.0, which implies a skewness value of 0.7. Henceforth we will use these alternative forms for f_w and f_θ rather than using the more complicated expressions (19) and (20). Thus, for simplicity we have chosen not to use the full formalism of the mass-flux approach. We will show later that the third moments resulting from use of the simpler functions (21) and (22) are reasonably consistent with the assumptions of the mass-flux approach. Note that (21) and (22) are not our approximations for w′² and θ′² throughout the entire PBL. In the following section we will give a complete derivation of these variances.

It is interesting to note that, using (21) and (22), w′²θ′ = (f_w/f_e)w′θ′² ∝ (w∗/θ∗)w′θ′² = (h/w∗)βgw′θ′², so that our parameterization implies that w′²θ′ is proportional to the buoyancy term in its budget equation. This is consistent with the result of Moeng and Wyngaard (1989), who find the buoyancy term to have the most important contribution relative to the other terms in the bottom-up component of the w′²θ′ budget equation. However, it neglects the mean-gradient contribution that is the major part of the top-down component. We will return to this question in section 3 below.

2) Parameterizing W ′q² and W ′³

Rather than using the mass-flux approach, the turbulent kinetic energy flux w′q² that appears in Eq. (4) is represented, following Therry and Lacarrère (1983). By using a third-order scheme and some experimental data, they simplify the equation for the kinetic energy flux, which for a horizontally homogeneous and Boussinesq approximated PBL, is given by

View Expanded

where U, S, E, and B represent production terms due to the horizontal velocities, the momentum fluxes, the TKE components, and the buoyancy, respectively; and P and M represent the pressure and the molecular term, respectively.

The assumptions of Therry and Lacarrère include

1. stationary equilibrium;

2. neglecting U, S, and M;

3. neglecting the horizontal velocity variance contributions to the buoyancy term B, which implies

   

   

   View Expanded

   
4. after rewriting E as

   

   

   View Expanded

   

   neglecting the terms that involve the gradient of the horizontal variances; and
5. assuming a “return-to-isotropy” for the pressure term

   

   

   View Expanded

   

   where c₃ = 8.

Therefore, the TKE flux can be written as

View Expanded

We use the expression given by Eqs. (19) and (21) to evaluate w′²θ′.

In order to evaluate the vertical velocity flux w′³, we note that it represents the major component of the w′q². In fact, the results from the Air Mass Transformation Experiment (AMTEX) as reported by Lenschow et al. (1980) show that about two-thirds of w′q² comes from the w′³ contribution. This is also consistent with Moeng and Wyngaard’s LES results. Therefore, we approximate w′³ as

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Here it must be noted that this approximation for w′³ is mainly valid for the convective boundary layer. In neutral and stable cases both w′q² and w′³ are small, and they are sometimes represented in terms of down-gradient diffusion approximation or neglected. Here we retain (29) for all cases.

3) The momentum flux transports

The turbulent fluxes of the two momentum fluxes w′²u′ and w′²υ′, which appear as diffusive terms in Eqs. (6) and (7), respectively, are not explicitly parameterized in this model. We assume that the diffusive terms in these equations are balanced by the buoyant production βgw′θ′. Therefore, with the assumption of quasi-steady state, Eqs. (6) and (7) become a balance between the pressure terms and the shear production terms:

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This balance is discussed by Therry and Lacarrère (1983), who depict the momentum flux budgets for the VOVES experiment. They also make the same assumption for their model. However, it must be noted that, although their results support the existence of such a balance in the lower part of the PBL, this is not so in the middle and upper part of the PBL. In fact, the transport term and the buoyancy term have the same sign in the middle and the upper part of the PBL. Therefore, a more accurate parameterization of the third-order moments in the momentum flux equation is needed. This will be investigated in future work.

b. The second-order moments

In this section we consider the second-order statistical moments. These include the heat flux w′θ′, the two momentum fluxes w′u′ and w′υ′, the vertical velocity variance w′², the temperature variance θ′², and the turbulent kinetic energy. We parameterize these quantities by making a steady-state assumption and by implementing the third-moment approximations described above.

1) The heat flux

With the steady-state assumption the heat flux equation given by Eq. (8) becomes

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The first term on the left-hand side of this equation represents the pressure covariance, which is parameterized as

View Expanded

where the first term, following Rotta (1951), is included as a return-to-isotropy contribution with the timescale τ, and the second term is a buoyant production term with c₄ = 0.5 as proposed by Moeng and Wyngaard (1986). The second term in Eq. (32) is the turbulent transport term, which is typically either neglected (Deardorff 1972a) or parameterized as a down-gradient diffusion. However, we use Eqs. (19) and (21) to represent this term. We define the quantity S as

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where η = (3/2)(z/h)^2/3. We assume S is a linear function of η. Then assuming quasi-steady state in the heat flux equation and assuming τ to be slowly varying in η, we solve the heat flux equation as discussed in appendix A (with an example of S in Fig. 1) to obtain the expression

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where

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with the quantities with subscript l representing the values computed at the top of the surface layer z = z_l = 0.1h; here η = η_t = 1.0 is the height at which S vanishes.

For convenience we write (35) as

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where

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and

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Note that K^loc_h is similar in form to the usual heat diffusivity based on local turbulence quantities. It is also interesting to compare our counter-gradient term γ_h with the previously proposed counter-gradient formulations of Deardorff (1972a), and Holtslag and Moeng (1991). The second term in (41) is just half of the γ_DE proposed by Deardorff, who neglected the transport term in the heat flux budget. By approximating the transport term as the pressure covariance term plus the constant term 2((w∗(w′θ′)_s)/h), Holtslag and Moeng obtain a counter-gradient term, γ_HM, that is given by

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In the central part of the convective PBL, for example, where z/h = 0.4, τ is of the order of h/2w∗ and E is is approximately 0.5, which implies that the first term in (41) is approximately one-third of γ_HM. The γ_hr term is new and depends mainly on the surface layer buoyancy and mean production. It is important to note that, although quantitatively similar, γ_DE and γ_HM have different physical interpretations. While the former results from the buoyancy production, the latter comes from the turbulent transport. Our result is a combination of the two.

2) The momentum flux

We have considered using several approaches in order to parameterize the momentum flux. One obvious approach is to carry out an analysis similar to that of the heat flux with appropriate counter-gradient terms; however, not enough experimental evidence exists to suggest a simple parameterization for the transport term in this case. Therefore, as discussed earlier, we assume that the transport of momentum flux approximately balances the buoyant production such that, with the steady-state assumption, the flux equations reduce to Eqs. (30) and (31). We parameterize the pressure terms in these equations using the return-to-isotropy formulation, which leads to the usual down-gradient formulation for momentum with diffusivity K_m,

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where K_m is proportional to the local heat diffusivity K^loc_h with a proportionality constant Pr, the Prandtl number that is computed from the surface layer matching as discussed in appendix B.

As we will see in the results discussed in section 3, this formulation is not fully adequate for the convective boundary layer. The LES results show that nonlocal processes are significant in the transport of momentum. We are presently investigating a way of including these processes in the momentum flux formulation.

3) The turbulent kinetic energy

The turbulent kinetic energy is the only second-order quantity that is determined prognostically, and its evolution is described by Eq. (4). Except for the viscous dissipation and the pressure transport terms, all other terms have been parameterized in the previous sections. The viscous dissipation is represented following Therry and Lacarrère as

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where c_ϵ is a constant and l is the characteristic dissipation length scale. We use the length scale proposed by Therry and Lacarrère, which takes into consideration the different behaviors according to the various surface stability conditions and thermal regimes. For unstable conditions it is given by

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and for neutral and stable conditions it is given by

View Expanded

where 1/l_s = 0 for a locally unstable stratification and

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for a locally stable stratification. Therry and Lacarrère chose the numerical constants c_ϵ and d₁–d₅ so that they would fit the AMTEX experimental data (see appendix C). They show that, in the limit of very convective, neutral, and stable conditions, Eqs. (46) and (47) have the appropriate behavior.

Finally we represent the pressure transport term following Lumley (1978) as

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where c₅ = 0.1. This parameterization was also used by Canuto et al. (1994) in their second-order closure model.

4) The vertical velocity and the temperature variances

In order to use the heat flux equation formulated earlier, we need to compute the vertical variance w′² and the temperature variance θ′² whose rate equations are given by Eqs. (9) and (10).

The pressure term in the w′² equation is modelled to include the effects of buoyancy–turbulence interactions as in Zeman and Lumley (1979):

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where c₆ = 7.0, c₇ = 0.3 and the last term involving w′p′ is parameterized using Eq. (49). Therefore, with these approximations, and assuming steady state, Eq. (9) yields the following approximation for the vertical velocity variance:

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where

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Consistent with the discussion leading to Eq. (29) above, the transport term (∂w′³)/(∂z) could be represented as 2/3(∂w′q²/∂z), which we further approximate using Eq. (4) ignoring the temporal variation of q². This gives

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Note that this does not mean that Eq. (4) has been replaced by its steady-state version. We have simply used this version to diagnostically evaluate the third term on the right-hand side of Eq. (51).

In order to parameterize the vertical temperature variance, we use Eqs. (20) and (22) for the w′θ′² term and we model the thermal dissipation ϵ_θ following André et al (1978) as

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where c₁₁ = 2.5. Then, assuming steady state with these approximations, Eq. (10) yields

View Expanded

where c₁₂ = 1/(c_ϵc₁₁).

Note that both w′² as given by (51) and and θ′² as given by (54) are consistent with the surface layer forms used in Eqs. (21) and (22).

c. The timescale τ

In order to model the timescale τ, which enters the heat flux equation through the pressure covariance term Π = −(1/ρ̄)θ′(∂p′/∂z) in Eq. (32), we assume that it is of the form

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where α is obtained by matching the expression for Π at the top of the surface layer as discussed in appendix B. This form is equivalent to the one used by Zeman and Lumley (1976) and is widely used to represent turbulent timescales.

d. The determination of the boundary layer height h

The implementation of the closures in this model requires us to calculate the boundary layer height h. We follow the approach of Troen and Mahrt (1986), who calculate h by solving

View Expanded

where Ri_cr is the critical bulk Richardson number assumed to be 0.5 in our model. The temperature θ_s for neutral and stable conditions is taken to be the surface temperature; however, for unstable conditions, θ_s is adjusted to represent the temperature of the convective thermals in the lowest part of the PBL as

View Expanded

where b = 8.58, z₁ is the lowest model level, and the velocity scale w_m is given by

w_mu³∗c_mw³∗^1/3(58)

where c_m = 0.6.

e. Summary of the boundary layer equations

Hereafter we designate the new boundary layer scheme that has been formulated above as the TKE model. In its one-dimensional version, this scheme includes the following main features.

• A nonlocal eddy diffusivity (K) for the buoyancy flux as given by Eq. (39);

• a counter-gradient term for the buoyancy flux as computed from Eq. (41);

• a prognostic equation for the TKE with a corrective buoyancy term in the TKE flux as in Eq. (28) and with Therry and Lacarrère’s dissipation length scale as given by Eqs. (46) and (47); and

• a down-gradient formulation for the momentum fluxes as given by Eqs. (43) and (44).

A complete list of the model equations is presented in appendix C.

a. Case I: Buoyancy-driven PBL

Let us first consider the case where the PBL is buoyancy dominated with relatively small shear. The depth of the PBL in this case is on the order of 1000 m with a convective velocity scale (w∗) of approximately 2 m s⁻¹, a friction velocity (u∗) of about 0.7 m s⁻¹, and where the ratio h/L is about −15. Figure 2 shows the initial mean variable profiles for this case. The potential temperature profile shows a strong capping inversion on the order of 6 K and the ū and the ῡ profiles show little shear at the top of the mixed layer.

In order to examine how well the model reproduces the temperature profiles and the potential temperature entrainment flux, we compute the following entrainment ratio,

View Expanded

where

View Expanded

and

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where t₀ and t_f are initial and final integration times, respectively. These integral quantities, which were used in Ayotte et al. (1996) for the evaluation of model performance of various PBL parameterizations, measure the extent to which entrainment contributes to the mixed layer averages of the mean variables. The computed value of R for the TKE model is 0.21, whereas the value from the LES model is 0.20. The close agreement between the models is a result of including the nonlocal effects in the model. In this strongly capped environment, the entrainment is determined by the complex interaction between the strong updrafts and the capping inversions, which depends on the surface heating and the shear across the entire mixed layer and the entrainment region. Therefore, nonlocal effects play a crucial role in determining the entrainment rate.

As shown in Fig. 3, where the final potential temperature and the heat flux profiles obtained from the TKE and the LES models are depicted, the TKE model is able to accurately reproduce the structure of the PBL, including the entrainment zone. The mean temperature profile is well mixed, which is typical of buoyancy-driven PBLs where the updrafts that are on the order of the PBL height cause nonlocal mixing. The heat flux is a linear function of height with negative values at the inversion layer that is associated with the entrainment of the overlying air. Consistent with the entrainment calculation, the downward heat flux at the inversion is about 20% of the upward surface heat flux.

The cross points in Figs. 2 and 3 are computed using a low vertical resolution similar to the third-generation Canadian Centre for Climate Modelling and Analysis GCM for which the new scheme is designed. The lowest level is at 60 m and the other levels have a uniform resolution of 200 m such that there are only 5 grid points in the convective PBL. Even at such a low vertical resolution, the model behaves well, and it is in good agreement with the higher resolution results and the LES output. Similar results were obtained for the other cases.

The normalized effective eddy diffusivity K_h is shown in Fig. 4, along with the K profile proposed by Holtslag and Moeng (1991) that is given by

View Expanded

where R_H is the ratio of entrainment flux to the surface heat flux. They obtain this formulation by curve fitting to the values of the LES data. They suggest R_H = −0.2, which is the value used in Fig. 4. The two curves agree well except near the top of the surface layer where the surface layer matching is carried out, causing K_h to vanish there.

For the purposes of comparison, we also present in Fig. 5 the nondimensional counter-gradient term of our formulation and that of Holtslag and Moeng, which is given by Eq. (42). In the central part of the PBL they both give a value of 5, which corresponds to the dimensional value of 0.6 × 10⁻³ K m⁻¹. This value is close to the original value proposed by Deardorff (1972a), which was 0.7 × 10⁻³ K m⁻¹.

The vertical velocity variance normalized by w²∗ is depicted in Fig. 6. Its profile exhibits the characteristic parabolic structure that peaks at z/h = 0.4 with a value of 0.4 (e.g., Lenschow et al. 1980). The peak for the TKE model is only slightly smaller than the LES peak and the overall agreement between the two models is excellent.

The temperature variance is shown in Fig. 7a. It decreases rapidly with height in the surface layer and becomes nearly constant in most of the mixed layer. There is a local maxima near the inversion level due to the large vertical inhomogeneity in the temperature that is caused by the large temperature jump in this part of the PBL.

For most of the boundary layer, the structure of this quantity agrees well with LES results. However, the maximum at the top of the boundary layer is substantially underestimated. This is due to the neglected mean-gradient contribution in the third-order parameterizations. This contribution consists of the major part of the top-down component, hence it can be improved by merging the expression in (54) with the top-down variance proposed by Moeng and Wyngaard (1989), which is given by

View Expanded

where c_t = 0.12 is computed by assuming a ratio of entrainment and surface flux of −0.2. As depicted in Fig. 7b, this merging, applied only to the top 20% of the PBL, gives a more pronounced peak at the inversion in better agreement with LES results. However, it has little effect on the vertical structure of the fluxes. Consequently it is not essential for the cloud-free boundary layer that is the focus of the present study.

The magnitude of the mean horizontal velocities ū and ῡ are shown in Fig. 8. While the ū profile is well mixed with a slight negative gradient in the LES model, in the TKE model it increases roughly linearly with height to its geostrophic value. On the other hand, while the ῡ profile is well mixed in the TKE model, it is a roughly linearly increasing function of height in the LES model. This discrepancy is a result of the down-gradient, assumption on the momentum fluxes. The LES model shows the flux w′u′ in Fig. 9a is linear and negative in the absence of a positive gradient in ū in the middle of the PBL, implying a counter-gradient process and nonlocal mixing. In order to support the negative flux as shown in the figure, the TKE model requires some shear since no counter-gradient processes are included for the momentum fluxes in this model. Similarly, for the ῡ velocity the negative gradient on the upper part of the PBL and the positive gradient in the lower part of the PBL are consistent with the down-gradient formulation of w′υ′ (Fig. 9b), which is positive in the upper part of the PBL and negative in the lower part of the PBL. Simple experiments with counter-gradient expressions indicate that the LES profiles can be reproduced by adding some counter-gradient terms in the momentum flux formulations. However, since we do not have a general enough counter-gradient formulation, we decided to keep the down-gradient formulation for now. We are presently investigating a way of incorporating the important processes in the momentum flux formulations.

The dimensionless profile of the sum of the three velocity variances, q² (twice the eddy kinetic energy), is presented in Fig. 10. Although the LES model shows more efficient mixing than the TKE model, the two curves are in general agreement. Since shear production is very small in this case, buoyant production along with turbulent transport and viscous dissipation determine the vertical distribution of energy. As Fig. 11 shows, the turbulent transport T = − (∂w′q′²)/(∂z) implies a loss of energy in the lower half of the PBL, and it implies a gain in the upper half. This is consistent with the LES result where the nondimensional TKE flux w′q′²/w³∗ increases with height from the surface to the central part of the PBL, where it reaches the maximum value of 0.1 and then decreases up to the top of the PBL (see Moeng and Wyngaard 1989). Buoyant production, on the other hand, implies energy gain, except in the top 20% of the PBL where there is a negative heat flux associated with the entrainment process.

b. Case II: Shear-driven PBL

We now consider the case with zero surface-flux and a strong shear. The PBL height h, in this case, is on the order of 500 m and the flow is strongly capped as in the previous case. Since there is no surface heat flux, both w∗ and θ∗ are zero, which, according to our parameterization of third-order terms, implies neglecting the turbulent transport terms in the heat flux and temperature variance equations and assuming a down-gradient transport for the turbulent kinetic energy. Therefore, the nonlocal characteristics of our parameterization are lost and all the turbulent fluxes and variances become down-gradient with a local eddy diffusivity K^loc that, in effect, reduces our parameterization to an equivalent of the level 2.5 Mellor–Yamada scheme (Mellor and Yamada 1974). The level 2.5 scheme is among a hierarchy of turbulent schemes developed by Mellor and Yamada, who used a systematic expansion procedure to analyze the departure of turbulent flows from the state of local isotropy, and it has been widely used in geophysical applications (Yamada 1977; Miyakoda et al. 1983; Gaspar et al. 1988; Crawford 1993). Although the level 2.5 model has a major deficiency in that it underestimates mixing and entrainment in the convective boundary layers, it has successfully been used to simulate the neutral and the stable boundary layers. Therefore we expect our model to perform well in these conditions as well.

As shown in Fig. 12, where the final profiles of the heat flux and the vertical velocity variance are depicted, good agreement is obtained between the LES and the TKE model. All the other turbulent quantities and the mean variables computed by the model are also in close agreement with the corresponding LES results.

c. Case III: Free-convective PBL

We now consider a free-convective PBL with the surface heating equal to Case I but with no geostrophic wind and u∗ = 0. As pointed out in Ayotte et al. (1996), most PBL models, even those with nonlocal and counter-gradient features substantially underpredict entrainment for this case. This may be due to the fact that fully nonlocal parameterizations ignore the partial local mixing that exists in the entrainment zone. On the other hand, single point models, such as the level 2.5 Mellor–Yamada, where the heat flux parameterization is based mainly on the local Richardson number, essentially ignore the nonlocal aspect of the mixed layer structure, resulting in weak entrainment.

The entrainment ratio R computed using Eq. (59) gives a value of 0.19 for our scheme, which is close to the LES computed value of 0.2. The heat flux profile and the profile of the vertical velocity variance for this case are also shown in Fig. 13. Although the vertical variance is slightly underpredicted in the central part of the PBL by the TKE model, the heat fluxes computed by the TKE model and the LES model are in agreement throughout the PBL. Similarly, Fig. 14 shows the normalized total variance q², which is well simulated by the model.

d. Skewness and third moments

As we have discussed in section 2, the use of the mass-flux concept leads to the parameterizations of the third-order moments w′²θ′ and w′θ′² in terms of the skewness, the heat flux, and the appropriate variance as given by Eqs. (19) and (20). Rather than computing the skewness and implementing the full expressions, we approximated the functions f_w = S_w (w′²)^1/2 and f_θ = S_w (θ′²)^1/2 by f_w = c₁ (z/h)^1/3 w∗ and f_θ = c₂θ∗, respectively. These approximations were chosen for their simplicity and because they are reasonably consistent with the LES results for third moments. Here we further examine these assumptions.

Figure 15 shows the skewness computed using the final profiles of w′³ and w′² computed a posteriori, from the model for the Case I simulation. Clearly the skewness is not a constant as was assumed. However, the vertically integrated average of the skewness turns out to be 0.7, which is the constant value assumed in the model. As we can see from the figure, 0.7 is a reasonable approximation of the skewness for the most of the PBL. Exceptions are in the surface layer where it is half this value and the entrainment zone where it is near unity.

Figure 16 depicts the third moments computed using Eqs. (19) and (20) with the skewness and the variances calculated a posteriori from the model and the third moments computed with the approximated functions given by Eqs. (20) and (21), as well as the third moments computed from the LES model. Both w′²θ′ and w′θ′² computed by the TKE model are in reasonable agreement with the LES model except near the top of the PBL where the approximated functions (20) and (21) are not valid. For w′θ′², the assumption of constant θ′² is certainly violated at the top of the PBL where it has a large jump. The moments computed with the full skewness expression are multiplied by a factor of 1.5 in the figure. It turns out that the full skewness expression gives third-order moments that are smaller than the LES and the TKE model by this factor. As we can see from the figure, the w′²θ′ computed a posteriori is, with this scaling factor included, in good agreement with results of both LES and the TKE model. However, this is less true for the structure of w′θ′². The gradient of w′θ′² computed by the full skewness expression is larger in the lower part of the PBL and smaller in the upper part than the value computed from the simplified expression. It is noteworthy that, although different from the LES result and the TKE model, the w′θ′² profile of the full skewness expression is not inconsistent with the Minnesota data (Kaimal et al. 1976).

Although this comparison indicates that the simplified assumptions that reduced the complex skewness expression to the more manageable form are reasonable, it does not reveal the full impact of using the complete nonlinear set of differential equations for the third-order moments. This is because the full nonlinear feedback is ignored in this a posteriori computation.