K-theory-based scheme (BKT)

There are several K-theory-based schemes using different similarity theories. Since these are somewhat similar formulations, we consider the formulations suggested by Businger et al. (1971) and Hass et al. (1991) to represent the turbulent processes in the surface layer and mixed layer. Previous studies (Chang et al. 1987; Hass et al. 1991) have indicated that this type of formulation can provide better representation of turbulent mixing processes than other K-theory schemes. The coefficient of vertical eddy diffusivity, K_z, for the surface layer is

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for the stable or neutral mixed layer; it is

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and for the unstable mixed layer, it is

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where k is the von Kármán constant, u_* is the friction velocity, h is the depth of the boundary layer, Φ_h is the nondimensional temperature profile, and w_* is the convective velocity. Note that the formulation for an unstable mixed layer (usually for daytime convective conditions) is very sensitive to the specification of the depth of the boundary layer, normally a poorly estimated variable. In the free atmosphere, turbulent mixing is parameterized using the formulation suggested by Blackadar (1979) in which vertical eddy diffusivities are functions of the Richardson number and wind shear in the vertical. This formulation can be written as

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where K_o is the background value (1 m² s⁻¹), S is the vertical wind shear, ι is the characteristic turbulent length scale (100 m), Rc is the critical Richardson number, and Ri is the Richardson number

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where g is the acceleration due to gravity and Θ_v is virtual potential temperature.

Turbulent kinetic energy scheme (TKE)

The prognostic equations used in this scheme to explicitly calculate the turbulent kinetic energy E and its dissipation rate ϵ are those suggested by Mellor and Yamada (1974). This scheme is often called a one-and-a-half-order closure scheme in which the unknown terms in the prognostic equations are parameterized in terms of local gradients of dynamic and thermodynamic parameters. The coefficient of vertical eddy diffusivity is calculated from the ratio of E and ϵ. Surface-layer similarity profiles (Businger et al. 1971) are used for obtaining boundary conditions for the prognostic equations for E and ϵ, while for the mixed layer the E–ϵ scheme is used. For further details, the reader is referred to Alapaty et al. (1994). In our study, we performed the model simulations using the second option described at the beginning of section 2 (i.e., the simulation domain is horizontally homogenous in each vertical layer). Thus, the advection terms in the governing equations (not shown) of E and ϵ have no effect on the model simulations. Therefore, these tests neglect the three-dimensional aspects of the TKE scheme (horizontal advection of E and ϵ), which are unique among these schemes. The coefficients of eddy diffusivity for momentum and heat can be written as

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where c₂ is an empirical constant (Detering and Etling 1985) and Φ_m and Φ_h are nondimensional functions for momentum and heat (Businger et al. 1971).

Transilient turbulence parameterization (TTP)

Turbulent eddies that exist in the ABL can transport heat, mass, and momentum over large vertical distances, often comparable to the depth of the ABL. Thus, at least during convective conditions, turbulent vertical mixing processes can be nonlocal. Stull and Driedonks (1987) use “transilient” (from the Latin for “jump over”) to indicate nonlocal vertical mixing in their parameterization of the ABL. Let S be any variable such as potential temperature, horizontal wind components, mixing ratio, or the concentration of a trace gas species in the atmosphere. The new value of S due to subgrid-scale turbulent vertical mixing for a grid cell i at a future time (t + Δt) can be written as

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where C is called the transilient matrix and i and j refer to two different grid cells in a column of atmosphere. In the turbulent mixing between grid cells i and j, C_ij represents the fraction of air mass ending in grid cell i that came from grid cell j, grid cell i is considered the “destination” cell, and grid cell j is considered the “source” cell. Thus, the change in the variable S due to subgrid-scale vertical mixing for grid cell i after a time interval Δt is a simple matrix multiplication with the value of S in the source cell. Estimation of the transilient matrix is the closure problem in this parameterization. This can be solved by considering the turbulent kinetic energy equation in nonlocal form. After parameterizing the unknown terms using the horizontal wind and temperature parameters, mixing potentials (Y) are obtained. Then, the transilient matrix C_ij can be written as

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where m_j is the mass of air in cell j, and ∥Y∥_∞ is called L_∞ norm of matrix Y. The diagonal elements (for i = j) of the transilient matrix can be written as

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Asymmetric convective model (ACM)

The prognostic equation for the asymmetric convective model is the same as that of any other nonlocal closure scheme. However, the ACM differs from the TTP in that the transilient matrix is greatly simplified for greater computational efficiency and in the methodology used to compute fractional mass mixing rates. The ACM is based on Blackadar’s nonlocal closure scheme (Blackadar 1979), which is based on the assumption that the turbulent mixing is isotropic (i.e., symmetric) in the ABL. Noting that the observational evidence and large-eddy simulation modeling studies of the mixing processes in a convective boundary layer are essentially asymmetric (i.e., turbulence is anisotropic) (Schumann 1989), Pleim and Chang (1992) modified this model by adding asymmetry in the vertical mixing processes. However, the ACM can be used only during convective conditions in the ABL. For other stability regimes, we use the BKT scheme described above to represent the ABL processes. Turbulent mixing in the ABL for any dynamic or thermodynamic variable can be written as

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where the elements in the matrix M represent mass mixing rates. This equation is similar to the transilient turbulence parameterization [Eq. (15)]. However, only a few pathways that represent the dominant mixing scales in the convective boundary layer are considered, resulting in a very sparse transilient matrix (see Fig. 1b) that can be numerically solved much faster than the TTP. Specifically, upward transport originates in the bottommost layer and goes to all layers above in the convective boundary layer. Downward transport goes from each layer to the next lower layer. This simulates rapid upward transport from the surface layer by buoyant plumes and more gradual compensatory subsidence. The calculation of the matrix elements is based on the conservation of sensible heat flux in the vertical direction. If Mu and Md represent upward mixing and downward mixing rates, respectively, then (16) can be rewritten as

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where Δσ is the relative mass in or thickness of cell i in a numerical model. The upward and downward mixing rates are estimated using the sensible heat flux. See Pleim and Chang (1992) for details and performance tests of the ACM.

The 11 July 1987 case study

During this day, the second IFC of the FIFE, clear sky conditions were observed until afternoon. Cloud-camera observations indicated overcast sky conditions after 1500 LT. Because the current version of our model does not consider processes related to cloud formation and its effects on atmospheric radiation and surface processes, model predictions after 1500 LT may not be comparable to observations. The temporal variations in the net radiation reaching the ground estimated by the model and from the observations are shown in Fig. 2a. Model estimation of the net radiation is very close to the observations up to 1530 LT; after that, the model estimates deviated because the model does not consider dynamic and thermodynamic processes related to cloud formation. Since the treatment of surface layer processes is the same in all four of the mixed-layer schemes, almost the same net radiation is estimated in all cases (differences among the results are about ±5 W m⁻²).

Turbulent latent heat fluxes in the model are estimated using the prognostic equations for soil moisture and parameterization of vegetative transpiration (Noilhan and Planton 1989; Jacquemin and Noilhan 1990; Pleim and Xiu 1995). Figure 2b compares these predictions with surface turbulent latent heat fluxes measured using the eddy correlation method. The predicted latent heat fluxes from the four ABL schemes differ from each other by about 0–30 W m⁻², even though surface processes are represented by a single set of formulations. These differences arise from the different representations of turbulent mixing processes in the four ABL schemes. In other words, turbulent mixing processes can affect surface fluxes by changing gradients across the surface atmosphere interface. Measured latent heat fluxes show a maximum of about 415 W m⁻² occurring around 1300 LT. Until 1400 LT, TTP-predicted latent heat fluxes are the closest to the observations. However, the differences between the schemes are well within the uncertainty of the measurements. After 1400 LT, predicted values from each of the mixed-layer schemes deviate greatly from the measurements due to cloud effects.

Figure 2c shows the temporal variation in the measured and predicted sensible heat fluxes from the four schemes. Until 1400 LT, predicted sensible heat fluxes from all of the schemes differ from measurements. Sensible heat fluxes are lowest with the TKE scheme and highest with the BKT scheme, with a maximum difference of about 50 W m⁻². This is the reverse of the trend in predicted latent heat fluxes (Fig. 2b), where the TKE scheme is highest and the BKT scheme is lowest.

In air quality modeling, accurate specification of near-surface air temperature is crucial. This is because biogenic emission rates, which can influence the formation of ozone in the lower troposphere, are strongly dependent on air temperature. Thus, differences in the predicted air temperatures can result in different estimates of biogenic emission rates. We found that the predicted ground temperature (T_g1) is highest with the TTP and lowest with the ACM, with a maximum difference of about 1.5 K. A consistent difference of about 1–2 K between measurements and predictions occurs because the measurement locations of the fluxes and soundings are different from the locations where ground temperatures are measured. Predicted air temperatures in layer 1 (∼11 m AGL) with the TKE scheme and the TTP are closer to each other and to observations. Similarly, predicted air temperatures with the BKT scheme and the ACM are very similar to each other and are about 2–3 K lower than those in the observations.

Figure 3a shows the vertical variation in “observed” virtual potential temperature Θ_v estimated from the observed temperature and water vapor mixing ratio for different observational periods. The presence of the superadiabatic lapse rate, a typical feature during the daytime near the surface, is evident starting from 0900 LT. The stronger vertical gradient (inversion) present at higher altitudes indicates the approximate altitude of the top of the mixed layer in each Θ_v profile. The Θ_v profiles from 1400 LT contain the influence of dynamic and thermodynamic processes related to the reported cloud activity. In the Θ_v profile at 1900 LT, it appears that the top of the mixed layer is pushed to an altitude (1900–2000 m) relatively higher than those in the other profiles. However, this feature cannot be simulated with the current model version.

Predicted Θ_v profiles from the four ABL schemes are shown for every 2 h from 0700 LT to 1900 LT in Figs. 3b–e. The vertical profiles at 0700 LT represent initial conditions in all four cases. In the BKT simulation (Fig. 3b), starting from 0900 LT the inversion is located at a lower altitude than in the observations (Fig. 3a). Also, all Θ_v profiles except at 1700 LT in the BKT show the presence of vertical gradients (weak unstable lapse rates) within the ABL, and the BKT scheme does not simulate well the superadiabatic lapse rates near the surface. However, the mean Θ_v of the predicted mixed layer (309.25 K) at 1700 LT is very close to the observed value at 1800 LT (309.5 K). Though observations indicated cloud formation after 1500 LT, predicted Θ_v profiles after 1500 LT are close to the observations. This may be due to the absence of warm- or cold-air advection to the observational site.

In the TKE simulations (Fig. 3c), gradual growth of the mixed layer can be seen from the Θ_v profiles starting from 0900 LT, similar to that in the observations. Unstable lapse rates near the surface indicate the presence of the surface layer. However, these unstable lapse rates are not as strong as in the observations. The vertically uniform predicted Θ_v above the surface layer indicates the presence of a well-mixed layer as found in the observations. The predicted mean Θ_v of the mixed layer at 1700 LT is about 0.5 K lower than the observed value. In the TTP simulations (Fig. 3d), the Θ_v predicted profiles up to 1300 LT show that the altitude of the inversion is still confined to about 800 m, similar to that in the BKT simulations. However, significant growth in the depth of the well-mixed layer is found after 1300 LT. The unstable lapse rates predicted in the surface layer are stronger than in the observations. The mean Θ_v of the mixed layer at 1700 LT is the same as in the observations. The Θ_v profiles in the ACM predictions (Fig. 3e) indicate relatively better growth of the mixed layer as compared to the BKT and TTP simulations. However, unstable lapse rates in the surface layer are very weak compared to observations. The mean Θ_v of the mixed layer at 1700 LT is similar to that in the observations at 1800 LT, with a well-mixed layer present throughout the simulation.

Prediction of stronger unstable lapse rates in the TTP may be due to underestimation of the magnitudes of mixing coefficients in the surface layer. It was also found that the negative sensible heat fluxes (due to entrainment processes) near the altitude of top of the ABL are smaller in the TTP than that in the TKE, particularly until 1300 LT. Hence, erosion of inversion near the altitude of top of ABL is weaker in the TTP as compared to that in the TKE. Thus, in the TTP simulations, neglecting vertical transport of turbulence, underestimating mixing coefficients in the surface layer, and simulating weak entrainment near the altitudes of top of the evolving ABL may have contributed to the slower growth of the ABL until 1300 LT. In a modeling study, Ayotte et al. (1996) compared the normalized turbulent fluxes obtained from a large eddy simulation model with those obtained from some local- and nonlocal-closure schemes used in general circulation models. In one of the several case studies, they also found that all of the ABL schemes (including the TTP) used in their one-dimensional model could not reproduce the entrainment fluxes as compared to those in the LES. Thus, differences in the predicted Θ_v among the simulations can be attributed in part to the differences in the entrainment fluxes.

In the BKT and ACM simulations, the altitude of the top of the mixed layer must be diagnosed from wind and temperature profiles. We use the bulk Richardson number criterion, similar to that used by Pleim and Xiu (1995) and suggested by Holtslag et al. (1990), in determining the depth of the ABL. On the other hand, specification of the depth of vertical mixing is not required in the TKE and TTP simulations. This is because in the absence of clear air turbulence, the turbulent kinetic energy becomes negligibly small (∼1.0 × 10⁻²⁰ m² s⁻²) in the TKE simulations near the top of the mixed layer. Similarly, in the TTP simulations off-diagonal elements in the transilient matrix become zero at altitudes near and above the top of the mixed layer. In the BKT simulations, the Θ_v profiles in the ABL indicate that the vertical mixing is underestimated in the mixed layer and overestimated in the surface layer. This is a typical characteristic of local-closure schemes in which vertical transport is proportional to vertical local gradients. Therefore, to achieve a well-mixed profile, vertical eddy diffusivity must go toward infinity. Further, the calculated altitude of the top of the mixed layer seems to be underestimated. In the ACM simulations, turbulent mixing rates in the ABL are determined by the magnitude of the sensible heat flux at the prespecified top of the surface layer and the potential temperature difference between layers 1 and 2 using an empirical formula. Therefore, the magnitude of the superadiabatic surface layer is dependent on the thickness of the lowest model layer. The main advantage of the ACM over the BKT in convective conditions is that the nonlocal mixing results in very uniform mixed layers without resorting to extraordinarily large eddy diffusivities or very short time steps in a numerical simulation.

Vertical profiles of observed horizontal winds (not shown) indicate the presence of a nocturnal jet in the stable boundary layer at 0700 LT. Thus, shear production of turbulence is probably an important factor in the growth of the ABL during the morning hours by causing mixing through the capping inversion. This process is accounted for in the TKE and the TTP simulations. While the mixing rates of the ACM and the BKT do not directly respond to vertical wind shear, the specification of the ABL top depends on the local wind shear across the interface of the inversion. Therefore, as these schemes are integrated in time, vertical wind shear at the top of the ABL increases the modeled depth of the ABL, thus including part of the capping inversion in a more rapidly mixing ABL. In this way the inversion is eroded over time allowing ABL growth. Clearly, as shown in Figs. 3 and 4, the morning ABL growth is slower for the BKT than the ACM because the local mixing in the stable entrainment zone is much less than the nonlocal convective mixing in the ACM.

In the absence of cloud formation and advection processes, the vertical distribution of trace gas species in the lower troposphere is largely controlled by ABL processes. The water vapor mixing ratio can be considered as a surrogate for the trace gas species (nonreactive) (just as latent heat fluxes at the surface can be viewed as a surrogate for surface emission fluxes). Thus, the temporal behavior and vertical distribution of the water vapor mixing ratio q_v can provide some insight on the probable behavior of a surface-emitted nonreactive trace gas species. Vertical profiles of observed q_v at different hours are shown in Fig. 4a. Except at 1400 and 1800 LT, q_v profiles indicate a uniform distribution within the mixed layer. Larger gradients can be seen in the surface layer. These gradients in the surface layer may be due to errors in the measurements in the surface layer (0–100 m) and/or to the presence of large latent heat fluxes (∼450 W m⁻²) as seen in Fig. 2b. The mean q_v inside the ABL changes from hour to hour, both increasing and decreasing, in ways that cannot be explained by vertical mixing and surface fluxes alone. Therefore, one-dimensional models cannot possibly replicate all of these observed features.

Vertical variation in q_v at different hours of simulation with the four schemes are shown in Figs. 4b–e. In the BKT simulations (Fig. 4b), vertical gradients of q_v are present in the ABL during the entire period of simulation as a result of weaker vertical mixing. For example, the q_v profile at 1500 LT varies from 16.5 g kg⁻¹ at the surface to 11 g kg⁻¹ at 1200-m altitude while the observations (Fig. 4a) indicate a uniform value of 16 g kg⁻¹ in the mixed layer (up to 1400-m altitude). Further, due to the availability of large latent heat fluxes and BKT’s slower growth of the mixed layer as compared to the observations, the mean q_v of the mixed layer increases for the period 0700–1200 LT. Turbulence in the free atmosphere is parameterized using Eq. (13) in the BKT and ACM simulations. Variations in the BKT-predicted q_v above the ABL (1500 m) are due to the free atmospheric turbulence and somewhat resemble those in the observations. In the TKE simulations (Fig. 4c), vertical variation in q_v indicates almost uniform profiles within the ABL except at 0900 LT. Rapid growth of the mixed layer (shown later in Fig. 5) at 0900 LT in the TKE simulations causes overprediction of q_v between the 700- and 1200-m altitudes. In the surface layer (0–100 m), the TKE simulations also show the presence of vertical gradients in q_v. However, these are weaker than in the observations. Above the ABL (1500 m), the TKE scheme did not simulate free atmospheric turbulence, resulting in no changes in q_v during the entire simulation period.

Vertical variation in q_v in the TTP simulations (Fig. 4d) indicates slower development of the ABL until 1300 LT. After 1300 LT, predicted q_v profiles also indicate a sudden growth of the ABL, which results in a decrease of the mean q_v in the ABL as compared to that at 1300 LT. In the surface layer, strong vertical gradients in q_v are predicted by the TTP and are very similar to the observations. Temporal variations in q_v in the free atmosphere (altitudes above the ABL) are also present in the TTP simulations. Vertical variation in q_v in the ACM simulations (Fig. 4e) indicates somewhat mixed results compared with the other simulations. Mean values of q_v in the ABL during the entire period of simulation (except at 1900 LT) are very similar in the ACM.

Observed eastward and northward wind velocity profiles (not shown) indicate the presence of remnants of a nocturnal jet in the lower altitudes in the initial conditions (0700 LT). Presence of vertical wind shear in the ABL (an indication of weaker vertical mixing) and slower growth of the mixed layer are found in the BKT simulations. The TKE simulations (not shown) indicate that the altitude of the maximum wind in the ABL (∼1000 m) is higher than in the BKT simulations. In the TTP simulations (not shown), vertical wind shear associated with the nocturnal jet is largely reduced by 0900 LT, similar to the TKE simulations. In the ACM simulations, the wind profiles are somewhat similar to those in the BKT simulations. In general, the effects of differing growths of the mixed layer in each of the simulations are evident in the predicted u and v wind profiles.

The depth of the ABL is a key input in air quality simulation models. Errors in its specification can significantly affect the predicted concentration fields. In the TKE scheme and the TTP the top of the ABL is diagnosed from a TKE criterion, while in the BKT scheme and ACM top of the ABL is specified from the temperature and wind profiles using the bulk Richardson number as described above. Therefore, accurate specification of the ABL height is crucial to these schemes since it determines the vertical extent of the convective mixing. Figure 5 shows the temporal evolution of the depth of the boundary layer. The bulk Richardson number criterion (Holtslag et al. 1990) is used to estimate the depth of the ABL in the BKT and ACM simulations. Since measured depths of the mixed layer are not available, the above methodology is also used to estimate the actual depth of the ABL (referred to as “OBS” in Fig. 5) using the observed winds and temperature profiles. Except for 1100 LT and 1300 LT, the TTP shows higher ABL depths than the other schemes do. In general, all schemes except the TTP predict lower values for ABL depth than the “observed” values. The maximum difference among the schemes for the predicted ABL depths is about 600 m, which translates to about 33% of the maximum predicted depth. Thus, variations in the predicted/estimated depths of the ABL may influence air quality simulation results.

The 6 June 1987 case study

During our second case study, the first IFC of the FIFE, clear sky conditions were observed during the entire period of simulation. Thus, comparing the measured net radiation with the model estimates will reveal errors in the radiation formulations, initial specification of surface, and atmospheric characteristics (e.g., albedo, emissivity). We found that the model estimates are very close to the observations. This indicates the accuracy of surface input data and the model formulation of radiative processes for clear sky conditions. Next we consider model prediction of turbulent sensible and latent heat fluxes. Figure 6a shows measured and predicted sensible heat fluxes. In general, peak values of predicted sensible heat fluxes in all simulations are higher than the measurements by about 15–40 W m⁻². Though the initial conditions and the formulation used to represent surface layer processes are the same in all of the simulations, the slight differences among the predictions indicate the effects of different representations of turbulent mixing processes. Analogous differences in the predicted latent heat fluxes (Fig. 6b) are smaller.

Of interest to meteorological and air quality simulation modelers is better prediction of near-surface winds, because these can affect the estimation of surface turbulent fluxes of thermodynamic parameters and chemical species. We compare surface observations of u and v winds with the model-predicted winds in the lowest layer (Figs. 7a and 7b). Since the lowest level in the model is located at about 10.9 m AGL, predicted winds can be directly compared to the measurements made at 10 m AGL. Except for the first 2 h of simulation, the trends in the predicted winds in all simulations follow the observations closely. On average, the TTP-predicted winds are closest to the observations, and the ACM predictions are the farthest from observations (overprediction).

We also compared predicted vertical variation in Θ_v with the observations at different hours (not shown). Compared with observations, superadiabatic lapse rates in the surface layer are lower in the BKT and ACM simulations, higher in the TTP simulations, and comparable in the TKE simulations. A feature associated only with the TKE simulations was the prediction of mixing in the residual boundary layer between 0700 and 0900 LT as observed. We also found that warming of the ABL (as indicated by the Θ_v profiles) is different in each of the simulations until 1300 LT, indicating the effects of different mixing schemes on the predicted ABL.

Predictions of the horizontal winds in a one-dimensional model are very sensitive to the prescribed geostrophic wind profiles. In all of our simulations we prescribe geostrophic wind profiles that were used by Pleim and Xiu (1995). Since horizontal pressure gradients are not readily available, they used a qualitative procedure to estimate temporally evolving geostrophic winds. Vertical variation in the u wind (eastward component) obtained from observations is shown in Fig. 8a. Again, the presence of a nocturnal jet at 0700 LT is evident from the u wind profile between the surface and 1000-m altitude; the core of the jet is located at about 500 m AGL. The u wind profile at 1600 LT shows the presence of a weaker vertical wind shear, indicating the effects of stronger turbulent mixing processes. Figures 8b–e show the predicted u wind profiles from the four ABL schemes. With time, all schemes except the TKE scheme show a gradual decrease in the vertical wind shear in the ABL, similar to the observations. With the TKE scheme, the coefficient of eddy diffusivity for momentum seems to be overestimated; as a result, vigorous vertical mixing leads to well-mixed u wind profiles from 0900 LT on. After 1300 LT, differences among the predicted u winds are about ±1 m s⁻¹ in all simulations.

Vertical variation in the v wind (northward component) is shown in Fig. 9a. The presence of the nocturnal jet is somewhat evident (0700 LT) in the lower altitudes of the atmosphere. Also note that the v winds are uniform with height for all of the observational periods. Predicted v wind profiles for the four schemes are shown in Figs. 9b–e. Again, with the TKE scheme vigorous vertical mixing results in a well-mixed v profile above 200-m altitude by 0900 LT. With the other schemes, the decrease in vertical shear in the v winds is more gradual in time, similar to the observations.

Figure 10 shows the temporal variation in the depth of the ABL. As before, observed winds and temperature data were used to estimate the probable altitude of the top of the ABL utilizing the diagnostic formulation suggested by Holtslag et al. (1990). During the morning period, all of the ABL schemes underestimate the “observed” ABL depths. Among the predicted depths of the ABL, the BKT depth is the lowest in comparison with the observations and the TTP depth is the highest. The maximum difference among the predicted ABL depths is about 250 m.