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Comparative Evaluation of Eddy Exchange Coefficients for Strong and Weak Wind Stable Boundary Layer Modeling MAITHILI SHARAN
AND
S. G. GOPALAKRISHNAN*
Centre for Atmospheric Sciences, Indian Institute of Technology–Delhi, New Delhi, India (Manuscript received 2 April 1996, in final form 26 August 1996) ABSTRACT Five local K-closure formulations and a TKE closure were incorporated in a one-dimensional version of the Pielke’s model, and a comparative evaluation of the closure schemes was made for strong and weak wind stable boundary layer (SBL). The Cabauw (Netherlands) and EPRI-Kincaid site (United States) observations were used for this purpose. The results indicate that for the strong wind case study, the profiles of turbulent diffusivities in terms of shape, depth of significant mixing, and the height above the surface where diffusion reaches a maximum are more or less the same for the various closure schemes. Only the magnitudes of mixing produced by various closure schemes are different. This difference produced by various closure formulations causes minor but noticeable changes in the mean wind field and thermodynamic structure of the model SBL. However, although the profiles of turbulent diffusivities become weak, variable, and poorly defined under weak wind conditions, the mean profiles become insensitive to the differences in the diffusion that arise due to various parameterization schemes. Apart from the TKE closure scheme, Estournel and Guedalia simple local closure scheme is able to produce the essential features of the SBL quite well.
1. Introduction The study of the stable boundary layer (SBL) is of vital importance in the field of air pollution modeling. Most of the industrial stacks are located within the SBL and the fate of the air pollutants released from the stacks is critically dependent on the structure of the SBL. The turbulent exchange processes are primarily responsible for shaping the structure of the SBL in terms of mean temperature, wind, and humidity profiles (Lacser and Arya 1986). Boussinesq (1877) proposed an analogy between molecular diffusion and turbulent diffusion to mathematically model the turbulent exchange processes in the planetary boundary layer (PBL), and according to it, the vertical turbulent fluxes are related to the mean vertical gradients by a turbulent exchange coefficient, KZ. The central problem associated with first-order and one-and-one-half-order closure modeling is to find suitable forms of the turbulent exchange coefficients that must account for the effects of turbulence. A number of parameterization schemes for the exchange coefficients have been developed. Yet there appears to
*Current affiliation: Department of Environmental Sciences, Cook College, Rutgers University, New Brunswick, New Jersey. Corresponding author address: Dr. Maithili Sharan, Centre for Atmospheric Sciences, Indian Institute of Technology–Delhi, Hauz Khas,110016 New Delhi, India. E-mail:
[email protected]
q 1997 American Meteorological Society
be no unique scheme for specifying the turbulent exchange coefficient in the PBL. The turbulent exchange coefficients in the surface layer based on Monin–Obukhov similarity theory (Monin and Yoglom 1971) work reasonably well for all stability classifications. However, above the surface layer, there is a considerable uncertainty regarding the magnitude and height dependence of the exchange coefficients. In order to model the turbulent processes in the PBL in a realistic manner, it is essential to consider the vertical variation of the exchange coefficients either explicitly or implicitly. In the explicit closure schemes, a profile is prescribed to account for the vertical variation of the turbulent exchange coefficients (O’Brien 1970; Businger and Arya 1974). The lower boundary condition required to specify such a profile is usually obtained on the basis of the Monin–Obukhov similarity theory. The O’Brien (1970) profile, which assumes that the turbulent exchange coefficient follows the cubic polynomial, simulates the convective conditions in the PBL fairly well (Pielke 1974; Yu 1977) and is presently being used in many of the mesoscale models like the Pielke’s model and the URBMET model (Shir and Bornstein 1977). After sunset, as the stability grows, the size of the turbulent eddies becomes small and with the increasing stability the eddy will no longer feel the effect of the surface. Under these conditions, Nieuwstadt (1984) pointed out that it is much more appropriate to use a local closure scheme (McNider and Pielke 1981; Es-
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tournel and Guedalia 1987) for the parameterization of the turbulent exchange coefficients. Further, such a scheme may be able to capture the turbulent characteristics aloft that have been found to influence the longrange transport of pollutants (McNider et al. 1988). In view of the above facts, local closures are being used in various mesoscale models (Pielke’s model, Penn State model, etc.) to parameterize the turbulent exchange processes in the SBL. In a local-K closure (i.e., implicit closure scheme), the turbulent exchange coefficients are expressed in terms of the local shear, mixing length, and the local stability function (Blackadar 1979) depending upon the gradient Richardson number. Apart from the use of simple first-order closure schemes to study the turbulent exchange processes in the boundary layer, several investigators (Wyngaard 1975; Andre et al. 1978; Blackadar 1979) have studied the turbulent features in the SBL by using higher-order closure schemes that involve explicit solution of the prediction equations for the boundary layer fluxes (Businger 1984). However, the requirement of enormous computational resources and a poor understanding of the complex equations for turbulence have limited the application of these schemes. As a compromise between simple first-order closure schemes that grossly account for turbulence and higherorder closure schemes that have very limited practical applications, the models based on turbulent kinetic energy (TKE) closure schemes are being used in the boundary layer studies (Holt and Raman 1988). Yu (1977) tested the performance of 14 parameterization schemes using the O’Neill fifth period and Wangara day 32 experiment data. The most satisfactory simulation in wind speed was reproduced by the TKE closure model, and the O’Brien’s (1970) K-formulation performs well under convective conditions. Holt and Raman (1988) made an exhaustive review and comparative evaluation of multilevel boundary layer parameterizations for first-order and TKE closure schemes. The performance of the schemes was tested using MONEX 79 data. They have shown that the mean structure of the boundary layer is fairly insensitive to the type of closure scheme, given that the scheme properly accounts for turbulent boundary layer mixing and that a TKE closure is preferable to first-order closure in predicting the overall turbulence structure of the boundary layer. Recently, Genon (1995) made a comparative evaluation of three closure schemes that included a simple firstorder closure formulation proposed by Louis (1979) and two TKE closure formulations. The performances of the schemes were tested for three experiments: the Wangara experiment for clear sky conditions, the Cabauw, and the Joint Air–Sea Interaction experiments concerning the interaction between turbulent and radiative processes in cloud layer (fog and stratocumulus). The diagnostic diffusion coefficient proposed by Louis (1979) was slightly modified to take into account the cloud
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layer and this gives satisfactory results in view of its simplicity. Almost all of the above studies have been restricted to strong–moderate wind situation. However, weak wind conditions occur frequently over the Tropics and it is a situation that has been less well understood (Sharan et al. 1995, 1996; Gopalakrishnan et al. 1997, submitted to J. Atmos. Sci.). In the present study, we wish to identify a closure scheme that reproduces the essential features of the SBL under strong and weak wind conditions. The turbulent parameterization schemes include five local closures with different local stability functions and a TKE–mixing length closure. The performances of the closure schemes were tested using the Cabauw (Netherlands) and EPRI-Kincaid site (United States) observations for the strong and weak wind conditions, respectively. As either of the experiments were conducted over a fairly flat terrain in the scale of 20 km 3 20 km, a one-dimensional model would be adequate to simulate the mean meteorological fields. 2. The boundary layer model A meteorological boundary layer model is used to study the evolution of the SBL. The model is a onedimensional version of a hydrostatic mesoscale meteorological model developed by Pielke (1974). Except for the parameterization of turbulent exchange coefficients, the model is the same as that discussed by McNider et al. (1988) and Sharan et al. (1995). a. Governing equations Equations for the horizontally homogeneous flow, potential temperature, and mean specific humidity are given by
1
2
]U ] ]U 5 f V 2 f Vg 1 KM ]t ]z ]Z
1 2 ] ]u 1 ]R ]u 5 1K 2 1 ]t ]Z ]Z rC 1]Z2
]V ] ]V 5 2 f U 1 f Ug 1 K ]t ]z M]Z H
(1)
(2)
(3)
p
1 2
]q ] ]q 5 KH , ]t ]Z ]Z
(4)
where Ug and Vg are the components of geostrophic wind, f is a Coriolis parameter, R is the net radiative flux, r is the density of air, and Cp is the specific heat capacity at constant pressure. In the present study, it has been assumed that KM 5 KH 5 KZ. For the parameterization of longwave fluxes in the SBL, Sasamori’s (1972) radiation scheme has been adopted. The shortwave radiation has been parameterized as discussed in Mahrer and Pielke (1977).
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b. Turbulent parameterization 1) FIRST-ORDER
CLOSURE PARAMETERIZATION
SCHEMES
The PBL is composed of two layers, namely the surface layer near the ground and the Ekman layer above it. In the surface layer, the turbulent exchange coefficients are calculated based on the similarity approach proposed by Businger et al. (1971). In the Ekman layer under convective conditions, the turbulent exchange coefficients are specified in terms of the O’Brien’s cubic polynomial. The depth of the convective boundary layer (CBL) is computed from Deardorff’s (1974) prognostic equation. While the growing CBL is prescribed by O’Brien’s profile, the turbulent exchange coefficients are parameterized in the SBL on the basis of the local approach (Estournel and Guedalia 1987) in terms of gradient Richardson number (Ri) and, accordingly, KZ 5 ,2SF(Ri),
F(Ri) 5 1.
(6)
Blackadar’s (1979) function. In this scheme, the local stability function is given by
5
F(Ri) 5
0 , Ri , Ri c
(7)
Ri $ Ri c ,
where Ric is the critical Richardson number (McNider and Pielke 1981). Louis (1979) function. In this scheme, F(Ri) is assumed to be an inverse function of Ri, that is, F(Ri) 5 (1 1 4.7 Ri)22, Ri . 0.
(8)
Estournel and Guedalia’s (1987) function. The stability function is defined as F(Ri) 5
5
(1 2 5 Ri) 2 , [(1 1 41 Ri)21 ]1.68 ,
0 , Ri , 0.16 Ri $ 0.16. (9)
Blackadar K-2 function (Bodin 1981). In this case, the variation of local stability function with Ri is assumed to be parabolic, that is, F(Ri) 5
5
b(1 2 a Ri)1/2 , 0,
KZ 5 (1 2 18 Ri)0.5,2S; Ri , 0 and (W9u9)0 . 0. (11) The mixing length , in Eqs. (5) and (11) is computed from the following relationship (Blackadar 1962; Estournel and Guedalia 1987):
(5)
where , is the length scale (mixing length) characterizing the energy containing eddies, S is the local wind shear, and F(Ri) is the local stability function of Ri whose functional form differs from one scheme to another. Here, we have used five different local stability functions and they are as follows: Neutral case. The turbulent exchange coefficient is taken to be independent of local stability parameter, that is,
1.1(Ri c 2 Ri) , Ri c 0,
Because two different kinds of exchange coefficients are used in the CBL and the SBL, a physically plausible transition must be made from profile to local scheme. The transition criteria used in the model is related to the sign of the surface heat flux. When the sign of the surface fluxes changes from positive to negative, the model switches from the profile to the local scheme. Even though the surface stabilizes, the decaying convective boundary layer aloft (also known as the residual layer) may contain superadiabatic lapse rates, at least during the initial period of transition. Under such conditions, when Ri is negative, the exchange coefficient in the residual layer has been parameterized as (Panosfsky et al. 1960)
0 , Ri , a21 Ri $ a21 ,
where a 5 1 and b 5 1 (Bodin 1981).
(10)
kZ
,5
1 2
,
(12)
kZ 11 l
where k is the von Ka´rma´n constant taken to be 0.35, Z is the height above the ground, and
l 5 2.7 3 1024
zGz , zfz
(13)
in which zGz 5 (U g2 1 V 2g )1/2 is the geostrophic speed. Equation (12) indicates that the mixing length increases linearly with height near the surface and approaches a constant value l far away from it. The parameter l depends on the geostrophic wind speed. As the mixing length [Eq. (12)] depends on geostrophic wind, this formulation may be more appropriate in dealing with strong and weak wind conditions. Apart from the first-order closure schemes discussed above, a TKE–mixing length closure scheme has also been adopted for this comparative study. We describe it here briefly for the sake of completeness. 2) THE TKE
CLOSURE
The temporal variation of TKE for a horizontally homogenous flow is given by (Holt and Raman 1988)
[1
2 1 2
]E ]U 5 Km ]t ]z
2
]V 1 ]z
]
2
1
1 2
1 2
g ]u ] ]E K 1 C Km 2 e, u H ]z ]z ]z (14)
where g is the acceleration due to gravity, C is a constant taken to be 1 in the present case, and e is the viscous energy dissipation defined as
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e5
(bE)3/2 , ,
(15)
where b is taken to be 0.2 (Holt and Raman 1988). The form of the exchange coefficients as used in the TKE closure scheme, based on the dimensional argument (Monin and Yaglom 1971), is given by KZ 5 C1,E1/2,
(16)
where C1 is a constant taken to be 0.4 (Holt and Raman 1988). There are many formulations for , available in the literature (Lacser and Arya 1986). This length scale can be determined on the basis of prognostic or diagnostic equations. However, Lacser and Arya (1986) pointed out that simple diagnostic relationships for mixing length work well in the SBL, and hence Eq. (12) has been used for the computation of ,. 3) INITIAL
AND BOUNDARY CONDITIONS
In order to solve the Eqs. (1), (2), (3), (4), and (15), initial conditions for the field variables and their values at the upper and lower boundaries are required. Initially the profiles of temperature and specific humidity were prescribed along with the geostrophic wind from Radiosonde observations. The initial condition for TKE does not seem critical (Yu 1977). We have used a value of 0.001 m2 s22 for the present study. The lower and upper boundary conditions required to solve the equations for the mean variables [Eqs. (1)– (4)] were obtained from the observations. The temperature at the surface at each time step was obtained by interpolating the available hourly observed temperatures at the surface. 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 held constant during the simulation. At the upper boundary, the geostrophic wind, the potential temperature, and the specific humidity were prescribed initially from the upper-air data and were held constant throughout the simulation. The lower boundary condition required to solve the Eq. (15) for TKE was obtained from the relationship (Holt and Raman 1988):
5
Z .0 L
3.75U*2 ,
E5
1 2
Z 3.75U* 1 0.2W * 1 2 L 2
2
2/3
U*2 ,
Z , 0, L (17)
where U* is the surface frictional velocity computed on the basis of similarity theory, L is the Monin–Obukhov scale length, and W* is the convective velocity (Mahrer and Pielke 1977). It was assumed that the TKE vanishes at the upper boundary. Equations (1), (2), (3), (4), and (14) have
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been solved numerically using a finite-difference approximation scheme (Pielke 1984). c. Observational data The measurements made during the night of 30–31 May 1978 at Cabauw were used for the strong wind case (Garratt 1982; Nieuwstadt 1985; Stull 1987; Tjemkes and Duynkerke 1989) study. The model was initialized with the observed profiles of temperature and specific humidity at 1000 UTC and the simulation was carried on for 20 h. At the lower boundary, the ground temperatures (Tjemkes and Duynkerke 1989) were prescribed. In 1980 and 1981, the Electric Power Research Institute (EPRI), USA, initiated a series of extensive field tests to examine the validity of existing dispersion models for concentration distributions up to 50 km downwind of the power plants and to provide databases for developing improved plume models. The initial phase of this project examined dispersion of the plume over a nearly flat terrain near the Kincaid site (EPRI report). A series of meteorological observations both near the surface as well as in the PBL were made in and around the Kincaid site. The observations of 26–27 May 1981 indicated that weak wind conditions persisted all through the night. For the case study of weak wind conditions, the above dataset has been used. The model was initialized at 1830 UTC with the observed profiles of temperature and specific humidity and the simulation was carried out for 14 h. Since hourly temperatures were only available at the screen height, the temperatures at the lower boundary were obtained by extrapolating the above observations on the basis of Monin–Obukhov similarity theory (Swati and Raghavan 1986). The list of input values required for the two simulations is given in Table 1. 3. Results and discussions Figure 1 shows the variation of the stability functions [Eqs. (6)–(10)] with the Ri. In the neutral limit, the function is independent of the Ri and is equal to 1. In the stable limit, the function vanishes. The Blackadar (1979) local stability function exhibits a linear variation, and the formulation of Estournel and Guedalia (1987) shows a behavior similar to that of Blackadar’s (1979) formulation. However, the formulation (Estournel and Guedalia 1987) approaches the stable limit much slower. Also, the value of the function is lower than that given by Blackadar. The Louis (1979) formulation exhibits an asymptotic behavior in Ri and the Blackadar K-2 formulation gives rise to a parabolic behavior. Further, the formulations (Estournel and Guedalia 1987; Blackadar 1979) produce a more stable nocturnal boundary layer than the one obtained from Louis (1979) and Blackadar K-2 formulations. It is to be mentioned that Estournel and Guedalia (1987) deduced the local stability function
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SHARAN AND GOPALAKRISHNAN TABLE 1. Values of the input parameters.
Parameters Albedo of land surface Roughness length Mean latitude Day of the year Time step Sea surface temperature Surface pressure Surface specific humidity Mean wind speed Mean wind direction Number of vertical levels Vertical levels
Cabauw run
EPRI run
0.3 0.15 m 52.08 30 May 1978 30 s 296.4 K 1000.00 mb 0.0027 10 m s21 1008 22 2, 15, 33, 57, 105, 151, 196, 253, 291, 373, 393, 447, 526, 677, 803, 1085, 1662, 1916, 2135, 2252, 2551, 3200 m
on the basis of experimental data under strongly stable conditions. Using the various closure formulations in the boundary layer model, now we analyze (a) diffusion characteristics, (b) thermodynamic structure, (c) wind fields, and (d) the surface-layer parameters under strong and weak wind conditions. a. Diffusion characteristics The profiles of turbulent exchange coefficients during various parts of the night are shown in Figs. 2a–d and 3a–d. For the strong wind case (Figs. 2a–d), there exist large differences in the magnitudes of mixing as produced by various closure schemes. The neutral K (not shown in the figure), Blackadar K-2, and the TKE formulations allow stronger mixing. The Estournel and Guedalia (1987) closure scheme produces the weakest mixing. Although the magnitudes of mixing are differ-
0.2 0.10 40.08 26 May 1981 30 s 298.39 K 1000.00 mb 0.0027 3 m s21 1058 22 2, 10, 25, 45, 80, 135, 225, 315, 405, 495, 585, 675, 765, 855, 945, 1035, 1125, 1215, 1305, 1485, 1575 m
ent for various closure schemes, the profiles of turbulent diffusivities in terms of the shape, height to which significant diffusion takes place, and height above the surface where diffusion reaches maximum are more or less the same for most of the closure formulations, except for the neutral K (not shown in the figure) and the Estournel and Guedalia (1987) schemes. The neutral K formulation produces large and uniform mixing throughout the SBL (not shown in the figure) and the Estournel and Guedalia (1987) closure scheme produces shallow mixing. Also, the height at which the maximum mixing occurs is less in this case. In the weak wind case (Figs. 3a–d) there exist large differences in the behavior of each of the formulations. The formulations of Blackadar (1979) and the Estournel and Guedalia (1987) produce SBL with very shallow mixing. The TKE closure allows stronger (in terms of magnitude) and larger (in terms of the depth) mixing. Louis (1979) closure produces significant mixing up to 50 m, with a well-defined maximum around 10 m. As the winds become weak (Figs. 3a–d), the depth of significant mixing, the magnitude of mixing, and the height of maximum mixing decreases. More importantly, the profile of turbulent diffusivities becomes weak, variable, and poorly defined. Also, a simple local closure, such as the Blackadar’s (1979) closure, that could produce salient features of diffusion as good as the TKE closure for the strong wind conditions (Figs. 2a–d) produces very weak and shallow diffusion under weak wind conditions (Figs. 3a–d). b. Thermodynamic structure
FIG. 1. Variation of the stability function [F(Ri)] with the Richardson number (Ri) for different local closure schemes.
For the strong wind case (Figs. 4a–d), the thermodynamic structures produced by all the closures are nearly the same and match quite well with the observations in the beginning of night. However, with the growth of stability, deviations arise in the performance of various closures. The differences in the magnitudes of diffusion produced by various closure schemes (Figs. 2a–d) result in minor but noticeable changes in the ther-
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FIG. 2. Profiles of the turbulent exchange coefficients simulated by different closure schemes at (a) 2000, (b) 2300, (c) 0200, and (d) 0500 UTC, under strong wind conditions.
modynamic structure that increase with the evolution of the SBL. The TKE closure performs the better in reproducing the overall temperature structure of the SBL (Figs. 4a–d). Although the deviations are large between the observations and the predictions, the structure of the more stable lower layer is reproduced much better by the closure scheme of Estournel and Guedalia (1987). For the weak wind conditions (Figs. 5a–c), although there exist large differences in the diffusivity profiles
(Figs. 3a–d), the mean temperature profiles are not very sensitive to the type of closure parameterization. Perhaps diffusion under weak wind condition is so small that the mean profiles become insensitive to the differences in diffusion that arise due to various parameterization schemes. Further, the temperature profiles are in fair agreement with the observations (Figs. 5a–c). Among the closure schemes that have been used in the present study, the
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FIG. 3. Profiles of the turbulent exchange coefficients simulated by different closure schemes at (a) 2100, (b) 2300, (c) 0100, and (d) 0300 UTC, under weak wind conditions.
TKE closure produces marginally better results, especially in the later part of the night (Fig. 5b). Early in the morning the neutral K formulation performs quite well (Fig. 5c). c. Wind fields Figures 6a–d and 7a–c represent the evolution of the mean wind fields in the SBL under strong and weak wind conditions.
In a convective boundary layer, a strong shear near the surface due to the effect of ground friction is usually observed (Sharan et al. 1995) in the wind profiles. Above the frictional layer, the wind profiles are nearly well mixed. At sunset a new shallow frictional boundary layer grows with a strong shear. This strongly sheared shallow frictional layer is well reproduced (Figs. 6a–d) by all the closure schemes for the strong wind case. Above the frictional layer, acceleration in velocity is observed as the wind fields adjust to a new balance with
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FIG. 4. Temperature profiles simulated by different closure schemes at (a) 2000, (b) 2300, (c) 0200, and (d) 0500 UTC, under strong wind conditions.
the existing pressure gradient (Blackadar 1957). This leads to the development of supergeostrophic component (Figs. 6a–d). For strong winds, as in the case of the thermodynamic structure, the wind fields (Figs. 6a–d) produced by all the closures are nearly the same. However, with the growth of stability, small but noticeable deviations arise in the performance of various closure schemes, es-
pecially in reproducing the wind maxima. The closure scheme of Estournel and Guedalia (1987) allows the development of a sharply defined nocturnal jet (Figs. 6b–d; see the right side of the panel). The height of the wind maxima is about 150 m and this agrees well with the observations (Stull and Driedonks 1987). The TKE, the Blackadar (1979), and the Louis (1979) closure schemes produce wind maxima at a slightly larger height
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FIG. 5. Temperature profiles simulated by different closure schemes at (a) 2100, (b) 0100, and (c) 0500 UTC, under weak wind conditions.
(Figs. 6b–d). Also, the wind maxima is not as sharp as that produced by the Estournel and Guedalia (1987) scheme. The development of the wind maxima depends on the mixing within the SBL. The scheme that allows least diffusion produces the sharpest wind maxima. On the other hand, the neutral K formulation allows large and uniform mixing throughout the SBL (Figs. 2a–d), and hence it is not able to reproduce the wind maxima well (Figs. 6a–d). The development of the frictional layer and the wind
maxima above this layer are restricted in the numerical models by the imposed geostrophic wind. For the weak wind case (Figs. 7a–c), all the closures produce a shallow frictional layer of about 50-m depth. Above this layer the wind maxima is not pronounced as clearly as in the strong wind case. Although there is a fairly good agreement between the observations and the predictions of the wind fields produced by all closures under weak wind conditions, none of the closure schemes is able to reproduce the
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FIG. 6. Wind profiles simulated by different closure schemes at (a) 2000, (b) 2300, (c) 0200, and (d) 0500 UTC, under strong wind conditions. Panel (i) includes Blackadar (1979), TKE, and Louis (1979) formulations. Panel (ii) includes Blackadar K-2, Neutral K, and Estournel and Guedalia (1987) formulations.
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FIG. 6. (Continued)
weak but distinct wind maxima correctly (Figs. 7a and 7b). The observations show that the wind attains a distinct maxima at a height much higher than that reproduced by various closure schemes. This may be due to the underestimation of the turbulent diffusivities under weak wind conditions (Figs. 3a–d).
We hasten to point out that the computed wind fields have been compared with observations in the weak wind case. The corresponding comparison was not feasible in the strong wind case due to nonavailability of data. However, the results obtained computationally are consistent with the studies of Stull and Driedonks (1987).
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FIG. 7. Wind profiles simulated by different closure schemes at (a) 2300, (b) 0100, and (c) 0300 UTC, under weak wind conditions. Panel (i) includes Blackadar (1979), TKE, and Louis (1979) formulations. Panel (ii) includes Blackadar K-2, Neutral K, and Estournel and Guedalia (1987) formulations.
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FIG. 7. (Continued)
d. Surface-layer parameters The computed surface-layer parameters such as U* and the heat flux have been compared with the observations in strong (Figs. 8a and 8b) and weak (Figs. 9a and 9b) conditions. The observed data show aperiodic fluctuations in surface-layer parameters. Some data presented by Gifford (1952), Kondo et al. (1978), and Nappo (1991) also show such behavior. These fluctuations may be attributed to the sporadic outbreaks of turbulence (Nappo 1991) and enhanced vertical diffusion of heat and momentum fluxes associated with the Richardson number instability (Blackadar 1979). The problem of sporadic breakdowns in the SBL is currently one of the challenging areas of research (Coulter 1990; Revelle 1993; McNider et al. 1995). However, it appears that none of the closure schemes, including the TKE closure, will be able to reproduce such turbulent episodes (Figs. 8a and 8b; Figs. 9a and 9b). It remains to be seen if a higher-order closure formulation will be able to give any better results. Since the computation of the eddy diffusivities in Eq. (5) depends on the Richardson number as well as the gradient of winds, some numerical experiments were carried out to find the sensitivity of the results on the vertical grid resolution. We increased the number of levels to 35 (with more resolution near the surface) and then we decreased the number of model levels to 16. The influence on the mean fields was insignificant for both strong as well as weak wind conditions. The dif-
fusion was affected with the decrease in the grid resolution, and the effect was more significant under weak wind conditions. However, the increase in grid resolution did not affect the diffusion. Hence it can be concluded that the present resolution is sufficient to study the structure of the SBL under weak and strong wind conditions. This is at least true for the present case study. Among the various local closure schemes discussed in this study, the Blackadar’s (1979) function was least affected by the changes in the grid resolution. This may be due to the fact that the critical Richardson number [Eq. (7)] used in this scheme depends on the grid resolution (McNider and Pielke 1981). 4. Conclusions In the present work, the performance of five local K formulations and a TKE, mixing length closure formulation were tested under strong and weak wind stable conditions. The Cabauw observations made on 30–31 May 1978 were used for the strong wind SBL and observations that were taken by the EPRI on 26–27 May 1981 at the Kincaid site were used in the case of weak wind SBL. Results indicate that for the strong wind case study, although the magnitudes of mixing are different for various closure schemes, the profiles of turbulent diffusivities in terms of the shape, height above the surface where diffusion reaches maximum, and height to which
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FIG. 8. Time evolution of (a) frictional velocity (b) kinematic heat fluxes simulated by various closure schemes under strong wind conditions.
significant diffusion takes place are more or less comparable between most of the closure schemes. The differences in the magnitudes of diffusion produced by various closure schemes cause minor but noticeable changes in the thermodynamic structure and wind fields that increase with the evolution of the SBL. However, although the profiles of turbulent diffusivities become weak, variable, and poorly defined under weak wind conditions, diffusion is so small that the mean profiles become insensitive to the differences in the diffusion that arise due to various parameterization schemes. Apart from the TKE closure scheme, the Estournel and Guedalia (1987) simple local closure scheme is able to produce the essential features of the SBL quite well. None of the closure schemes used in this study, including the TKE closure, are able to reproduce the intermittency observed in the data in strong as well as weak wind conditions. It remains to be seen if a higherorder closure model would be able to simulate the surface fluxes better. Although strong stability is generally more associated with weak wind conditions, it is not always so. The
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FIG. 9. Time evolution of (a) frictional velocity and (b) kinematic heat fluxes simulated by various closure schemes under weak wind conditions.
present study, for example, indicates that although the winds were strong on 30–31 May 1978 at Cabauw station, a strongly stable boundary layer prevailed, especially during the later part of night. Hence, in contrast to the more widely used classifications of weakly, moderately, and strongly stable boundary layer, in dealing with the mesoscale diffusion problems in the SBL it appears that a classification based on winds is more appropriate. However, it should be noted that general conclusions cannot be drawn on the basis of a single study, and hence more observational programs are required to update our knowledge of the SBL, especially under weak wind conditions. Acknowledgments. The authors are thankful to Prof. R. T. McNider for providing us with the meteorological model. Thanks are due to Prof. M. P. Singh for his valuable comments on this study. The authors also wish to acknowledge Dr. Stephen Tjemkes for providing us with the Cabauw data.
MAY 1997
SHARAN AND GOPALAKRISHNAN REFERENCES
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