Model equations

The following is the set of equations that govern a horizontally homogenous flow in the boundary layer:

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where u and υ are the velocity components of the wind along the east–west and north–south directions, respectively; u_g and υ_g are the components of the geostrophic wind; θ is the potential temperature; q is the specific humidity; f is the Coriolis parameter; K_M and K_H are, respectively, the vertical exchange coefficients of momentum and heat, related to the turbulence kinetic energy E by the Monin and Yaglom (1971) dimensional argument (i.e., K_M = K_H = 0.4ℓE^1/2); g is the acceleration due to gravity; and C is a constant taken to be 1. The mixing length is parameterized by 1/ℓ = 1/(kz) + 1/λ, where k is the von Kármán constant, z is vertical height, and λ = 0.000 27G/|f|, in which G is the geostrophic wind speed (Blackadar 1962; Sharan and Gopalakrishnan 1997). The source term F_θ includes contributions from short- and longwave heating/cooling processes and is expressed as

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in which R_N is net radiative flux, ρ is the density of air, and C_p is the specific heat capacity at constant pressure.

Equations (1)–(5) were approximated by a version of the Crank–Nicholson scheme (Pielke 2002). Instead of the usual equal weighing at the current time steps, the new time step is overweighed. In this study, the beta factor has been taken as 0.75 (Paegle et al. 1976).

Radiation parameterization

The vertical structure of the PBL is driven by the diurnal heating and cooling of the atmosphere. In the absence of phase change of water substances, the shortwave and longwave radiative fluxes contribute to the local heating and cooling processes. The shortwave radiation parameterization is discussed by Mahrer and Pielke (1977). For the parameterization of longwave fluxes in the NBL, Sasamori's (1972) radiation scheme has been adopted by many authors for its simplicity and computational efficiency (e.g., McNider and Pielke 1981). Quite often, as demonstrated later in this work, the contribution of longwave fluxes to the local heat budget is expected to be significant, and, as a consequence, a better scheme to parameterize the longwave radiative fluxes may be required. In this study, a scheme based on the emissivity approach (Mahrer and Pielke 1977) in which water vapor and carbon dioxide (CO₂) are considered as principal emitters of longwave radiation is adopted.

For a given temperature T, in kelvin, the radiative cooling at each layer i is computed from

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where the downward (R_d) and upward (R_u) fluxes for each layer j are given by

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in which σ is the Stefan–Boltzmann constant, subscripts m and 0 correspond to the top of the model and the surface, respectively, and ε_i,j is the emissivity matrix. The emissivity at the surface ε₀ is taken to be 1.

The computed radiative cooling at each time step provides F_θ from (6) after converting the temperature to potential temperature. Note that if an isothermal atmosphere is assumed and the radiative flux divergence is computed at a given level i, (8) and (9) reduce to the conventional form of Sasamori's scheme.

Determination of the inversion layer depth

In an ideal fair-weather nocturnal boundary layer, the inversion depth is determined as the height above the surface at which the local potential temperature gradient Γ vanishes. For computational purposes, it is taken to be the height where Γ is approximately equal to a preassigned small value δ. In the current simulation, two consecutive levels (z_i−1 and z_i) have been determined such that Γ_i−1 > δ and Γ_i < δ, in which Γ_i−1 and Γ_i are, respectively, the potential temperature gradients at the levels z_i−1 and z_i. This condition implies that there exists a distinct difference in stratifications between the NBL and the residual boundary layer above it and that the potential temperature in the residual layer is nearly dry adiabatic.

The inversion depth at which Γ is equal to δ is computed from

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The value of δ is taken to be 3.5 × 10⁻³ K m⁻¹, as used by André and Mahrt (1982).

Initial and boundary conditions

To solve (1)–(5), initial conditions for the field variables and their values at the upper and lower boundaries are required. In all four cases (Table 2), observed temperature profiles obtained from TSONDE soundings at the Kincaid site were used to initialize the model up to a height of 1.7 km. However, because there were no observations available beyond that height, upper-air soundings obtained from Peoria, Illinois, (≈80 km north of Kincaid), and Salem, Illinois, (≈100 km south of Kincaid) were interpolated and then used up to the model top, which was fixed at 5.7 km. The dewpoint temperature profiles for the initialization were obtained by interpolating the data from Peoria and Salem in all of the cases. The reanalyzed initial profiles of temperature and dewpoint temperature for the four simulations are depicted in Figs. 1a,b. The initial condition for TKE was set to 0.001 m² s⁻², which is not critical (Yu 1977). In all of the cases except WW1, the model is initialized with the observed data. In the WW1 case, the initialized profile at 1430 LST is chosen in such a way that the model is able to reproduce the observed temperature profile at 1830 LST. In the absence of observations, such an approach has been used in some of the studies on the convective boundary layer using First International Satellite Land Surface Climatology Project Field Experiment data (Avissar et al. 1998).

The lower and upper boundary conditions required to solve (1)–(4) for the mean variables were obtained from the observations. The temperature at the surface at each time step was obtained by interpolating the hourly surface temperatures. A no-slip condition was assumed for the components of velocity at the ground. The surface specific humidity at the lower boundary was prescribed initially from the observations and was held constant during the simulation. At the upper boundary, the geostrophic wind, temperature, and specific humidity were prescribed initially from the upper-air data and were held constant throughout the simulation; TKE was set to zero. The surface albedo of 0.2 and the surface roughness of 10 cm are considered. The surface specific humidity is taken as 0.0273 kg kg⁻¹. In the earlier study (Sharan and Gopalakrishnan 1997), the same value was used in the numerical simulations. However, it was indicated as 0.002 73 kg kg⁻¹ incorrectly. The time step used for integration is 30 s, the model is run with 22 levels in the vertical up to a height of 5764 m, and the mean latitude is taken as 40°N. The values of input parameters used in the numerical simulations are given in Table 3.

The lower boundary condition required to solve (5) for TKE was obtained from the relationship (Holt and Raman 1988)

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where u∗ is the surface frictional velocity, w∗ is the convective velocity scale, and L is the Obukhov length.

Mean structure of the NBL

Figure 2 shows the observed and simulated mean thermodynamic structure of the NBL at 0000 (top panel) and 0600 LST (bottom panel) for the simulations SW1, SW2, MW1, and WW1. In general, there is a good agreement between the model simulations and observations. Although the best results are produced for the weak-wind case (simulation WW1, Fig. 2d) with an error of 0.4 K in potential temperature, local warm air advection, especially at the upper levels around 0600 LST, was not captured by the model. As a consequence, the potential temperatures were underestimated by about 2 K in simulation SW1 (Fig. 2a) and by about 2.5 K in simulation SW2 (Fig. 2b) at 0600 LST. Nevertheless, with the decrease in the ambient geostrophic forcing, the simulations show a shallower NBL; this result is supported well by observations. Indeed, an estimate of the modeled inversion depths (Fig. 3) illustrates a systematic decrease in the inversion depths with weakening of winds. The stable boundary layer (SBL) height based on TKE (the height at which TKE is reduced to 5% of its surface value) is found to be within the inversion layer during the night in strong, as well as weak, wind conditions. The SBL height reduces in magnitude with weakening of winds (Fig. 3).

An analysis of the modeled cooling profiles (only the curves corresponding to 2200 and 0200 LST are given in Fig. 4, for the sake of representation) obtained in simulations SW1, SW2, MW1, and WW1 illustrates that in all of the cases the NBL develops into a three-layer structure, within the inversion layer (Fig. 3): 1) Very close to the surface, radiative cooling dominated over turbulence cooling. In this layer, late in the night, cooling from radiation may be offset by turbulence warming, causing net reduction in cooling. 2) A layer above it, cooling from turbulence dominates. 3) Near the top of the turbulent layer and above it, clear-air radiative cooling is the dominating mechanism. These findings are consistent with those of Garratt and Brost (1981), Tjemkes and Duynkerke (1989), and Gopalakrishnan et al. (1998). However, a comparison between SW1 (Fig. 4a) and MW1 (Fig. 4c) shows that while the depth of the layer cooled by turbulence is about 120 m or more in SW1 it is about 80 m under moderate wind conditions. In simulation WW1 (Fig. 4d), the depth of turbulence cooling is confined to a very small depth on the order of about 20 m. It appears that the geostrophic forcing has a very strong influence on these turbulence cooling profiles. The total cooling rates (K day⁻¹) from the model at the 100-m level from midnight to morning are 9.24, 9.88, 6.72, and 3.28 in the SW1, SW2, MW1, and WW1 cases, respectively. The corresponding total cooling rates (K day⁻¹) from the observations are 10.14, 13.27, 8.13, and 2.85. We have taken a 100-m reference because this is the level at which one could potentially notice the presence or absence of turbulence, depending on the strong- and weak-wind cases.

Figure 5 depicts the observed and simulated mean wind components (u and υ) for the simulations SW1, SW2, MW1, and WW1. Again, the best results are produced for the weak-wind case (simulation WW1, Fig. 5d). Simulation SW1 shows reasonable agreement with the observations, especially late in the night (Fig. 5a). A comparison of the weak-wind case (Fig. 5d) with the strong-wind cases (Figs. 5a,b) demonstrates that the u and υ components attain maximum value at much lower altitudes under weak wind conditions. Note that a clearly developed wind maximum, which is observed as well as simulated by the model well above 200-m altitude under strong wind conditions (Figs. 5a,b: bottom panels), is located at an altitude of less than 100 m for the weak-wind case (Fig. 5d, bottom panel). Indeed, a look at modeled eddy diffusivity K profiles (Fig. 6) sometime during evening transition and at later hours demonstrates that a sharp decrease in diffusivities has resulted in wind maxima. Note that as the mixing becomes weak and shallow (Fig. 6d), the wind maximum appears nearer to the surface. Also, a shallow and weak diffusion in the NBL implies restricted and weak turbulence cooling, as evidenced in Fig. 4d. Note that the comparison between the computed and observed wind components is not as good in the MW1 case as in the other three cases, because the observed data are disturbed. The computed eddy diffusivity profiles in Fig. 6 are dependent on the choice of specification of the mixing length.

The well-pronounced low-level wind maximum in all of the cases may be due to the occurrence of inertial oscillations over homogenous terrain. Blackadar (1957) assumed that the ageostrophic component of the wind, released of all frictional constraint near sunset, undergoes an inertial oscillation (IO) that leads to supergeostrophic values of the wind several hours later. In the NBL, Singh et al. (1993) assumed that the IOs may be produced as a result of the horizontal momentum released because of the deviation from the geostrophic wind (which is termed the ageostrophic wind) in the decaying boundary layer. The ageostrophic wind at sunset thus triggers the occurrence of IOs during the nighttime, which results in the low-level wind maximum or nocturnal jet (Stull 1988; McNider et al. 1993; Singh et al. 1993, 1997).

Figure 7 illustrates the model-predicted hodographs at three different heights for the simulations SW1, SW2, MW1, and WW1. The wind is observed to be supergeostrophic with a magnitude of 11.2 m s⁻¹ (simulation SW1, Fig. 7a) and 10.6 m s⁻¹ (simulation SW2, Fig. 7b) at 315 m, about 5–6 h after sunset in strong-wind cases. The maximum wind speed is found to be matching with the observed low-level wind maximum (Figs. 5a,b). In a similar way, supergeostrophic wind is observed at 160 m in moderate wind (simulation MW1, Fig. 7c) and at 50 m in the weak-wind case (simulation WW1, Fig. 7d). The model is able to produce the inertial oscillations. The height of the low-level wind maximum is found to be almost the same as that observed (Fig. 5). Note that the oscillations are produced at lower altitudes for the weak-wind case (Fig. 7d).

The components of mean wind during the night are given by (Stull 1988)

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where u_g is the geostrophic wind assumed along the x direction at sunset, F_uday and F_υday represent the departure of the winds from geostrophy at sunset, f is the Coriolis parameter, and t is the time elapsed from sunset. Equations (13)–(14) can be used for computing the geostrophic departures, F_uday and F_υday, at sunset for producing the IOs from the components of wind during the night at time t. We have computed the ageostrophic wind using the model-predicted u and υ components. The computed values of the geostrophic departure (m s⁻¹) are F_uday = 0.67 and F_υday = 2.07 for SW1, F_uday = 0.25 and F_υday = 1.64 for SW2, F_uday = 0.008 and F_υday = 0.414 for MW1, F_uday = −0.57 and F_υday = 0.56 for WW1. The amplitude of the IOs in general decreases as the wind becomes weak; however, in the MW1 case, it is slightly less than that of WW1.

Evolution of the lower NBL

Figure 8 illustrates the evolution of the observed and simulated temperatures at 2-, 10-, 50-, and 100-m altitudes for the SW1, SW2, MW1, and WW1 simulations. The observations and the simulations both show a cooling trend that diminishes with height in all of the cases. A comparison between observations and simulations shows that the agreement is better at lower levels than at higher levels (50 and 100 m). The maximum error between the predicted and observed values at 100 m in the SW1, SW2, MW1, and WW1 simulations are, respectively, about 2.5 (Fig. 8a), 2.8, 2.8, and 2 K.

Figure 9 shows the evolution of the simulated and observed winds at four levels from tower observations in all four of the simulations. In general, the observations show a lot of scatter, mainly because of coexistence of waves along with turbulence. The observations (Figs. 9b–d) clearly show oscillations in wind with a period of about 6 h, which is much smaller than the period 2π/f of an inertial oscillation. Although, as seen earlier (Fig. 7), the model is able to produce larger inertial oscillations, it is incapable of reproducing smaller oscillations on the order of about 6–7 h. At this time, even very high resolution, nonhydrostatic boundary layer models, such as large-eddy models, have only been marginally successful in reproducing some features of waves in the NBL. It is not expected for the simple model used in the current study to simulate these undulations. Nevertheless, because our major concern was the mean structure of the NBL, we compared the means of the observed wind structure with those simulated. In SW1 (Fig. 9a), the model overpredicts winds at 10 and 30 m, whereas at higher levels the simulated winds are reasonably close to the mean of the observations. Similar behavior of the simulated winds is observed in SW2 (Fig. 9b) also, except for the underprediction at 100 m (Fig. 9b). The simulated winds from the model are reasonably close to the mean of the observations in the WW1 (Fig. 9d) and MW1 (Fig. 9c) case studies, except for the overprediction at the 10-m level in the latter case.

In the NBL, wind shear is the driving mechanism that causes the transfer of turbulent heat and momentum fluxes. The heat flux very near the surface is largely influenced by two factors, namely, the wind near the surface and the difference in temperatures between the cooling surface and the air near the surface. Because the direct eddy correlation measurements are not available at Kincaid, the surface layer parameters, such as the friction velocity and surface heat flux, are calculated using the Monin–Obukhov similarity theory with Beljaars and Holtslag (1991) stability functions for momentum (ψ_m) and heat (ψ_h). The surface friction velocity and the heat flux are computed using similarity theory (flux-profile relations) and the observations in analogy with the model levels. The model uses the computed wind at the lowest level (2 m), the surface temperature, and the computed temperature at the second level (10 m) for calculating the surface fluxes at the height taken as the middle of the first two levels. The same levels are used in computing surface fluxes using observations.

Figure 10 shows the evolution of heat flux near the surface in simulations SW1, SW2, MW1, and WW1. In general, observations as well as simulations show almost a clear decrease in the magnitude of heat fluxes with weakening of winds. Whereas, for instance, the average heat flux in SW1 is about −20 W m⁻² (Fig. 10a), it is from −2 to −1 W m⁻² in WW1 (Fig. 10d). Heat flux computed from the model is comparable to that based on the corresponding observations (Figs. 10a–d). The observed heat fluxes for the moderate-wind case are larger than for the two strong-wind cases (Fig. 10). It may be observed from Fig. 9 that the mean wind near the surface (at 10 m) in the moderate-wind case varies from 1 to 4 m s⁻¹ whereas in the strong-wind cases it varies from 2 to 3 m s⁻¹. The observed heat fluxes in the moderate-wind case are higher because of the slightly higher observed surface wind speed and higher observed temperature difference between the first two levels (Fig. 8). However, in the strong-wind cases, the observed surface wind speed and the temperature differences are relatively small.

Derbyshire (1990) has given a formula for computing the magnitude of the maximum buoyancy flux B_max = R_fG²|f|/3, where R_f is the flux Richardson number and G is the geostrophic wind speed. This may be converted into the surface heat flux (W m⁻²) by multiplying B_max by a factor of ρC_pT/g (S. H. Derbyshire 2001, personal communication), in which T is taken as the surface temperature. Following Derbyshire (1990), a value of 0.25 is taken for R_f. For the sake of comparison, the magnitude of the maximum surface heat flux from the computations as well as the observations is presented. It is found that the maximum surface heat flux is 38 and 40 W m⁻² from Derbyshire's formulation, 26 and 27 W m⁻² from the model (Figs. 10a,b), and 28 and 40 W m⁻² from observations in the SW1 and SW2 cases, respectively. In the MW1 case, the maximum surface heat flux is 16 W m⁻² from Derbyshire's formulation, 15 W m⁻² from the model (Fig. 10c), and 58 W m⁻² from the observations. In the weak-wind case, it is 4.4 W m⁻² from Derbyshire, 2.8 W m⁻² from the model (Fig. 10d), and 6 W m⁻² from the observations. It is found that the maximum surface heat fluxes from the model are lower than those from Derbyshire's maximum in strong as well as weak wind conditions. However the maximum surface heat flux from the model is close to that of Derbyshire's maximum in the moderate-wind case. The Derbyshire's maximum flux is closer to the observed maximum in the SW2 case, whereas it is higher in the SW1 case and lower in the MW1 and WW1 cases when compared with the corresponding observed maximum.

Figure 11 illustrates the evolution of surface frictional velocity u∗ in simulations SW1, SW2, MW1, and WW1. The frictional velocity based on the observations exhibits aperiodic fluctuations in conjunction with the observed winds (Figs. 9a–d). Despite the presence of waves, the model is able to reproduce to some extent the mean evolution of u∗ in the SW1 and SW2 simulations. Also, the simulations indicate that the u∗ values should decrease from strong to weak wind conditions. However, it is not obvious from the observational data that u∗ decreases with the weakening of the winds. On the other hand, earlier studies (Estournel and Guedalia 1987) with Cabauw observations for strong wind and Valladolid, Spain, observations for weak wind show that frictional velocity and kinematic heat flux are smaller under weak wind than under strong wind conditions. Further, the calculated results obtained are in conformity with the earlier studies (Estournel and Guedalia 1987; Gopalakrishnan et al. 1998).

Integrated cooling budget within the inversion layer

We examine the integrated cooling budget within the surface inversion layer. The individual contribution of turbulent and radiative coolings within the inversion layer of depth h is obtained by integrating the thermodynamic energy equation (3), taking into consideration that the turbulent heat flux vanishes at the top of the inversion layer. We get

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where (w′θ′)|₀ is the turbulent heat flux at the surface, R_N|_h is the net radiative flux at h, and R_N|₀ is the corresponding flux at the surface.

The first term on the right-hand side of (15) represents the integrated turbulent cooling, and the second term gives the radiative cooling. It may be noted that the heat flux at the surface contributes to the turbulent cooling.

In a dry atmosphere, the contribution of radiative fluxes within the inversion layer is not expected to vary from strong to weak winds (Figs. 4a,d). However, the turbulent heat flux (Fig. 10) at the surface and inversion depth (Fig. 3) varies appreciably from strong to weak winds. When the wind becomes strong, the contribution of the heat flux to the cooling budget is large and, hence, the turbulent cooling mechanism dominates the integrated budget (Fig. 12a). On the other hand, when the wind becomes weak, the contribution of the turbulent heat flux to the cooling budget is small and, thus, the contribution of radiative cooling dominates the net cooling budget within the inversion layer (Fig. 12d). These findings are consistent with our earlier study (Gopalakrishnan et al. 1998).

Shear and buoyancy budget

Similar to the integrated cooling budget, we have analyzed the shear and buoyancy budgets in the NBL up to a height h_t at which TKE is 5% of that at the surface. For this purpose, (5) is rewritten as

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where β refers to the sum of diffusion and dissipation terms in the TKE equation (5).

Integrating (16) with respect to z from 0 to h_t, we get

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Relations (18) and (19) refer to the shear production and buoyancy production terms, respectively. The integrated effect of diffusion and dissipation terms is β_I, and −B indicates the buoyancy consumption term. Here, we are looking for the relative contribution of the shear and buoyancy to the total TKE. Integrals in (18) and (19) are evaluated numerically (Press et al. 1986).

Figures 13 and 14 illustrate the individual contribution of shear production and buoyancy consumption terms to the total TKE under different wind conditions. The shear production is found to be dominating over the buoyancy consumption in the strong (Figs. 13a,b) and moderate (Fig. 14a) wind conditions, whereas the contributions from shear production and buoyancy consumption are comparable in the growth of the NBL in weak wind conditions (Fig. 14b). The contribution from shear and buoyancy terms to the total TKE is consistent with the role of turbulence (Figs. 10–12) in the growth of NBL under strong and weak wind conditions. The flux Richardson number R_f is relatively smaller in the strong wind conditions (SW1, Fig. 13a) than in the weak wind conditions (WW1, Fig. 14b). The R_f is plotted for a representative case for each of strong and weak wind conditions for the brevity of presentation.

Discussion on assumptions of the model and availability of data

Now we discuss some of the assumptions taken in the model and availability of data for simulation purposes.

1. The one-dimensional boundary layer model used in this study assumes that the flow is horizontally homogenous and, thus, that the effect of advection is neglected. As mentioned in section 2, we have chosen four nights out of 250 days for which the influence of advective cooling on the evolution of NBL is minimum. Nieuwstadt and Driedonks (1979) have suggested that the differences between simulations and measurements may be attributed to the advective cooling. We are initiating the studies of analyzing the role of advection using a three-dimensional model. The criteria used in selecting the cases required that the temperature differences between observations at 0000 and 0600 LST at 850 hPa were less than 1 K, which implies weak large-scale advection, but there could still be relatively large advection occurring at regional and local scales.

2. High-resolution observations from the tethered balloon sounding at Kincaid were only available up to a height of about 1.65 km. Because no other upper-air observations were available at this station, the upper-air data were reanalyzed on the basis of data available at Peoria and Salem for heights beyond 1.65 km.

3. The boundary layer model used in this study assumes the persistency of geostrophic flow during the period of simulation. The model requires improvements to consider the nonstationary geostrophic wind.

4. Direct eddy correlation measurements were not available in the EPRI dataset at Kincaid; the surface-layer parameters (frictional velocity and heat flux) are computed using the surface-layer similarity theory and the tower observations for purposes of comparing them with those obtained from the model. However, the direct eddy correlation measurements will be useful for providing further confidence in the results. At this time, very limited observations are available in the NBL, especially under weak wind conditions. It is necessary to update the conclusions as and when extensive meteorological measurements, including turbulence data, become available under strong and weak wind conditions.

5. The hourly surface temperatures are prescribed externally from observations. Further, the surface temperature at each time step is obtained by linearly interpolating the temperature at two consecutive hours. It is proposed to consider the surface energy balance for computing the surface temperatures. The Kincaid site consists of agricultural fields. Therefore, the influence of vegetation cover on the surface temperatures remains to be seen by using surface energy balance, taking into consideration the vegetation layer/canopy.

6. The observations have shown aperiodic fluctuations in the case studies undertaken. Several datasets, for instance, Cabauw (Nieuwstadt and Driedonks 1979) and Wangara (Clarke et al. 1971), exhibit intermittency in the NBL. It appears that simple one-dimensional numerical models such as the one used in this study will not be able to reproduce these fluctuations (McNider et al. 1995). It remains to be seen whether a more complex, higher-order closure scheme for turbulence would be able to simulate these fluctuations reasonably well.

7. The mixing-length parameterization used in this study in turn depends on the geostrophic wind speed. Such a choice of mixing length is more appropriate here because it accounts for variation from strong to weak wind conditions. It is well known that the mixing length, or the large-eddy length scale, decreases with increasing stability. However, we are in the process of examining the sensitivity of various mixing-length schemes (Lascer and Arya 1986; Estournel and Guedalia 1987; Saiki et al. 2000) available in literature that are 1) independent and 2) dependent on stability. The study reveals that the conclusions from the current study remain qualitatively unchanged under strong-wind as well as weak-wind stable conditions, even when a stability-dependent scheme is used for determination of mixing length.