Evaluation of an Atmospheric Boundary Layer Model Used for Air ...

DECEMBER 1998

LIU AND LEUNG

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Evaluation of an Atmospheric Boundary Layer Model Used for Air Pollution Studies C. H. LIU

AND

D. Y. C. LEUNG

Department of Mechanical Engineering, University of Hong Kong, Hong Kong, China (Manuscript received 16 July 1997, in final form 5 January 1998) ABSTRACT A three-dimensional mesoscale meteorological model was developed based on second-moment closure equations that were solved by the finite-element method. This paper aims to evaluate the performance of the model under flat terrain and horizontally homogeneous atmospheric boundary layer conditions. The one-dimensional version of this model was tested against field measurements, a water tank experiment, and another numerical model. It showed several interesting behaviors of the atmospheric boundary layer under stable and unstable flows that are of primary interest for environmental studies.

1. Introduction Dispersion of passive contaminants in the atmospheric boundary layer is a controversial issue in air pollution studies because of its complexity and the important consequences it may bring to our environment. The transport phenomenon of contaminants in the atmosphere, which is a function of the source conditions, meteorological parameters, geographical locations, etc., is difficult to predict accurately. Atmospheric boundary layer simulation models are essential tools to analyze and understand the phenomenon of atmospheric pollutant dispersion. In the past 15 to 20 years, many numerical studies have been undertaken focusing on the dynamics of the atmospheric boundary layer. The most frequently used models can be classified into the following four categories: 1) First-order closure or k models (Pielke 1974; Cotton and Tripoli 1978), 2) Second-order closure models (Mellor 1973; Enger 1983), 3) Third-order closure models (Andre et al. 1978; Briere 1987), and 4) Large-eddy simulation models (Deardorff 1972; Moeng 1984). First-order closure models make use of the flux-gradient equation to represent the turbulent contribution. However, it is doubtful that the vertical turbulent vari-

Corresponding author address: Dr. D. Y. C. Leung, Department of Mechanical Engineering, 7/F, Haking Wong Building, University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail: [email protected]

q 1998 American Meteorological Society

ance w9w9 can be calculated accurately (Yang 1991). Large-eddy simulation models explicitly calculate the large eddies and parameterize the small eddies. These models give a fairly accurate simulation but are computationally expensive. Third-order closure models are basically for research only because of their large computation loads. Comparatively, meso-g-scale second-order closure models, which cover roughly the interval of 2–20 km, are less computer intensive and are good alternatives for the studies of atmospheric boundary layer. Most of the mesoscale boundary layer models used the finite difference approach for their computation. In contrast, use of the finite-element method is not so common despite its having several distinct advantages over the finite difference method. An important advantage is, unlike the finite difference method, its ability to use an unstructured grid, making it suitable to handle any arbitrary and complex geometries with ease. Pielke and Martin (1981), McNider and Pielke (1984), Enger (1990), and Koracˇin and Enger (1994) used the finite difference method to model mesoscale atmospheric boundary layer. The terrain is treated by using terrainfollowing coordinates. In deriving the transformation, the slope of the terrain is limited to 458, which need not be applied if the finite-element method is used. The boundary of any arbitrary geometry can be modeled by the concept of an isoparametric finite element (Zienkiewicz 1971) in which the same parametric function is used for interpolating the spatial variation of dependent variables within an element as well as describing the geometry. In addition, the finite-element method is very flexible in handling variable resolution meshes, and it is helpful in obtaining local refinement of resolution in regions of strong gradient. This flexibility makes the finite-element method particularly suitable for studying

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atmospheric boundary layer flow over complex terrain even with slopes greater than 458. Another essential characteristic of the finite-element approach is that the Neumann (differential)-type boundary conditions are incorporated naturally into the weighted residual formulation of the model (Dick 1996). Furthermore, it looks for a solution of some integral form of partial differential equations instead of the partial differential equations by integration, which makes it more accurate in reproducing the conservation laws (Ferziger 1981). More detailed historical reviews, development, and advantages of the finite-element method in meteorological studies can be found in Sasaki and Chang (1982). In the present study a unique three-dimensional, meso-g-scale, second-order closure atmospheric boundary layer model is developed for studying the air pollution problem. Due to the desire of handling complex terrains, the finite-element method is used. As a stepping stone for higher-level validation, one-dimensional runs for convective and stable flow were simulated and the results were compared with the following results from literature: the Wangara experiment (Clarke et al. 1971), the numerical model (Mellor and Yamada 1974), and the water tank experiment (Willis and Deardorff 1974). The following sections will discuss the mathematical model with a description of the numerical scheme and finite-element approximation. 2. Mathematical model a. Mean equations A three-dimensional, mesoscale meteorological model is developed (Liu 1998). The mean equations of motion (see appendix for nomenclature) in one-dimensional form under dry atmosphere are ]u ]p ] 5 2 1 (2u9w9) 1 f y , (1) ]t ]x ]z ]y ]p ] 5 2 1 (2y 9w9) 2 fu, ]t ]y ]z

and

1

2

1 ]p ]p (U g , V g ) 5 2 , , f ]y ]x (1) and (2) can be rewritten as ]u ] 5 f y 2 f V g 1 (2u9w9) ]t ]z

The turbulence closure scheme used is based on the model suggested by Andreń (1990), which is basically a ‘‘Yamada–Mellor 2.5 level’’ model but also features a parameterization of wall effects on the redistributive pressure terms. In this type of closure, all equations for second-order turbulent quantities, except turbulent kinetic energy, are reduced to analytical expressions. The turbulent kinetic energy is calculated by the following prognostic expression suggested by Enger (1990):

1

2 1

]q 2 ] 5 ]q 2 ]u ]y 5 a1 ql 1 2 2u9w9 2 y 9w9 ]t ]z 3 ]z ]z ]z q3 . B1l

1 2bgw9u9 2 2

(7)

c. Algebraic reduction Mellor and Yamada (1974) showed that the vertical momentum fluxes (u9w9, y 9w9) and vertical heat flux (w9u9) can be simplified further. After considerable algebraic manipulation, (u9w9, y 9w9) and (w9u9) can be reduced to (8)

]u , ]z

(9)

]u ]y

and 2w9u9 5 k h

where k m and k h are obtained from analytical expressions suggested by Enger (1990). Substituting (8) and (9) into (5), (6), and (3), the following equations can be obtained:

1 2 ]y ] ]y 5 2 fu 1 f U 1 1k , ]t ]z ]z 2 ]u ] ]u 5 f y 2 f Vg 1 k , ]t ]z m ]z m

(10)

(11)

and

1 2

]u ] ]u 5 kh . ]t ]z ]z (6)

1]z , ]z 2

(2u9w9 , 2y 9w9) 5 k m

g

(5)

2

The second and third term on the right-hand side of (7) are the wind shear and buoyancy production terms, respectively. Once the values of q 2 are obtained, the components of Reynolds stress and heat flux can be calculated with the use of mean values (u, y , u) and turbulent length scale l. The diagnostic equations suggested by Andreń (1990) are adopted for calculating the turbulence.

(4)

and ]y ] 5 2 fu 1 f U g 1 (2y 9w9). ]t ]z

b. Turbulence closure scheme

(2)

]u ] 5 (2w9u9). (3) ]t ]z Since the kinematic pressure gradient (]p/]x, ]p/]y) can be reduced in the form of geostrophic wind defined as

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(12)

Thus, the mathematical model can be expressed in terms

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LIU AND LEUNG TABLE 1. Case study data. Case 1 Wangara experiment (Clarke et al. 1971)

Case 2 Mellor and Yamada (1974)

Geostrophic wind U g and V g

Obtained from measurement of thermal wind and parabolic profile fitting

Kept at constant values of U g 5 18 m s21 and V g 5 0 m s21

Coriolis parameter f

20.826 3 1024 s21

0.88 3 1024 s21

Surface roughness z 0

1 cm

5 cm

of (7), (10), (11), and (12), which would be solved by the finite-element method to be discussed in section 4.

i 5 a1 z 1 a2 ln

1az 2 ,

(15)

3

d. Turbulent length scale The following expression for the turbulent length scale l proposed by Mellor and Yamada (1974), which is obtained from neutral turbulent flow data, is adopted:

l5

kz , kz 11 l0

where

E E

i z 5 a1 (z max ) 1 a2 ln

(13)

1za 2 . min

(16)

3

qz dz .

`

max

and 1 5 a1 (z min ) 1 a2 ln

0

1za 2 3

`

l0 5 (0.1)

where a 3 is a constant set equal to z 0 . The constants a1 and a 2 are related to the fixed number of grid points (i z ) in the z direction and the height of the computational domain at ground surface (zmin ) and top of boundary (zmax ) by the following two simultaneous equations:

(14)

q dz

0

A value of 0.35 is assigned to the von Ka´rmań constant k (Businger et al. 1971). It can be observed from (13) that l → kz as z → 0 and l → l 0 as z → `. The turbulent length scale given by (13) and (14) can be applied to both stably and neutrally stratified atmosphere (Enger 1990). Sun and Ogura (1980) stated that the typical values of the turbulent length scale within the convective boundary layer are between 100 and 600 m, which is more than twice that calculated from (13) and (14). Thus, they cannot be extended to unstably stratified atmosphere. e. Computation domain and grid system The height of the computation domain is chosen to be 2000 m for validating against Wangara experiment (Clarke et al. 1971) and 4000 m for comparing with numerical model of Mellor and Yamada (1974). These heights are large enough for the mesoscale perturbation to be well within the model domain and also for applying the Neumann boundary conditions along the top of the model. In this part of the validation, a one-dimensional calculation is performed. To improve the accuracy of the finite-element approximation, a log–linearly spaced vertical grid is adopted. The distribution of vertical grids can be described as follows:

Equation (16) can be solved for a1 and a 2 by Newton’s method (Hoffman 1993). Once the values of a1 and a 2 are determined, the vertical grid distribution can be obtained by (15). 3. Initial and boundary conditions Two cases are simulated for comparing the present model with the Wangara experiment (Clarke et al. 1971) and the numerical model of Mellor and Yamada (1974). Data of the geostrophic wind, Coriolis parameter, and surface roughness used in the above two cases of calculation are summarized in Table 1. The geostrophic wind is obtained by parabolic fitting of the thermal wind and surface geostrophic wind data. It should be noted that the surface geostrophic wind components are available hourly, while thermal wind data is normally available twice daily. Thus error may be found in the calculation of geostrophic wind. In case 1, the integration is started at 0900 LST on day 33 of the Wangara field observation. The initial profile of mean variables u, y , and u are obtained by curve fitting the data points of the Wangara experiment. In case 2, the initial potential temperature is set according to the following profile: u (z)| t50 285 K,  5 285 K 1 0.0035 K m21   3 (z 2 1000 m),

0 , z , 1000 m, z . 1000 m. (17)

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This case is a neutral flow up to 1000 m and is covered by a stable layer of temperature gradients 0.0035 K m21 above. The initial velocity is computed according to the steady-state equation of a neutral flow expressed by (17). In both cases, initial value of twice the turbulent kinetic energy q 2 is not available, so it is generated by the model itself. The initial condition of q 2 is firstly obtained by fixing the temperature and velocity profile. After 2 h of integration, an initial value for q 2 is obtained, which is then used as the initial profile of the turbulent kinetic energy. Close to the ground surface neutral flow relations are assumed with the following logarithmic velocity profile:

12

(18)

u t 5 [(2u9w90 ) 2 1 (2y 9w90 ) 2 ]1/4 .

(19)

(u, y )| z→0 ; (cosa, sina)

ut z ln , k z0

where a is the angle of the stresses vector on the ground surface and u t is the ground surface friction velocity, which is expressed as Here, a is assumed equal to constant when deriving the boundary conditions of velocity calculation. The following expression, suggested by Mellor and Yamada (1974), is used as the boundary condition of q 2 on the ground surface: q | z50 5 6.10u . 2

(20)

2 t

It should be noted that the ratio q /u 5 6.10 is a repeatedly observed surface value (Mellor 1973). Equation (18) is used to described the velocity profile near the ground surface and cannot be applied directly at z 5 0. For the first two grid points above the ground surface, z 2 . z1 . 0 (18) yields 2

2 t

]q 2 5 0, ]z

(22)

]u ]y 5 5 0, ]z ]z

]u 5 0.0035 K m21 , ]z

(23) for case 1

The finite-element method is used for the present study. The continuum is divided into many nonoverlapping small elements that are generally unstructured. The solution is approximated by a linear shape function. The governing differential equations in each element are then transformed into the corresponding finite-element equations by using variational principles or the weighted residual method. The finite-element equations are collected together to form a global system of algebraic equations. The nodal values of the dependent variables are then determined from this system of equations with proper boundary conditions. The above method is used to solve (7), (10), (11), and (12). The three-dimensional transport equations of u, y , u, and q 2 in the original three-dimensional model is solved by Petrov–Galerkin finite-element method (Liu 1998). Due to the removal of the advection terms in the transport equations, the present one-dimensional run can be solved by Galerkin finite-element method. Discretization of the mathematical model is discussed in the following section. The weak formulation of (7), (10), (11), and (12) can be written as follows:

E[

W

V

] E ] E ] R [ 1 2]

]u ]W ]u 1 km 2 W( f y ) dV 5 ]t ]z ]z

[W(2 f V g )] dV,

V

(25)

E[

W

V

]y ]W ]y 1 km 1 W( fu) dV 5 ]t ]z ]z

[W( f U g )] dV,

V

(26)

E[

(21)

Equation (21) provides the boundary conditions for the velocity calculation. Ground surface potential temperature u 0 is a Dirichlet boundary condition of the observed ground-level potential temperature. The Dirichlet boundary condition is a specification of the dependent variables along the boundary (Anderson 1996). At the upper boundary, the following boundary conditions are used for the integration:

]u 5 0.001 K m21 , ]z

4. Finite-element approximation

W

ln(z1 /z 0 ) (u1 , y 1 ) 5 (u2 , y 2 ) . ln(z2 /z 0 )

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V

]W ]u ]u 1 kh dV 5 ]t ]z ]z

W kh

G

]u n dG, ]z z (27)

and

E[

W

V

5

]

1 2

]q 2 5 ]W ]q 2 2q 3 1 a1 ql 1W dV ]t 3 ]z ]z B1l

E5 [ 1

W 2 2u9w9

V

2

]u ]y 2 y 9w9 1 2bgw9u9 ]z ]z

]6

dV, (28)

where W is the weighting function, V is the open connected domain, G is the boundary, and n z is the z component of the unit normal vector to G. The spatial derivative of the dependent variable f (u, y , u, and q 2 ) is discretized using the linear isoparametric element:

and

O N (z)f (t), n

f (z, t) 5

i

i

(29)

i

for case 2.

(24)

where N i(z) is a linear shape function. The weighting

DECEMBER 1998

function W is chosen to be equal to the shape function N. Therefore, (25)–(28) can be discretized and linearized in space as follows:

E[

]u

N

]t

V

E[

1 kc m

]N ]u ]z ]z

V

[M11 ] 5 [M22 ] 5 [K11 ] 5

E

] ]

2 N( f y ) dV 5

[N(2 f V g )] dV, [K12 ] 5

(30)

E

N

[K21 ] 5

[N( f U g )] dV,

V

V

2

(39)

6

][N ] T ][N ] kc dV, m ]z ]z

(40)

{2[N ] T [N ] f } dV,

(41)

{[N ] T [N ] f } dV,

(42)

6

(43)

V

R[ 1

]u c ]N ]u 1 kh dV 5 ]t ]z ]z

{[N ] T [N ]} dV,

V

(31)

E1

E

V

E5 E E E5 E V

V

]y c ]N ]y 1 km 2 N( fu) dV 5 ]t ]z ]z

N

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[K22 ] 5

]

2

]u N c k h n z dG, ]z (32)

G

][N ] T ][N ] kc dV, m ]z ]z

V

{F1 } 5

{[N ] T (2 f V g )} dV,

(44)

{[N ] T ( f U g )} dV.

(45)

V

and

E[ V

and

]

1 2

]q 2 5 ]N ]q 2 2qˆq 2 N 1 a1 qˆlˆ 1N dV ]t 3 ]z ]z B1lˆ

5

E5 [ 1 V

d ]yˆ 2gw9 u9 d ]uˆ 2 yd N 2 2u9w9 9w9 1 ]z ]z uˆ

2

{F2 } 5

]6

V

dV. (33)

This choice of the shape and weighting functions leads to the following semidiscretized Galerkin form of equation in matrix notation:

5 6

]f [M] 1 [K]{f} 5 {F}. ]t

[ ] [ ] M11

M22

,

K K12 [K] 5 11 , K21 K22 {F} 5

56

F1 , F2

where

56

u , y

{f} 5 {u},

(46)

E5 V

(34)

6

][N ] T ][N ] c kh dV, ]z ]z

(47)

26 d G.

(48)

and

(35)

{F} 5

R5 G

1

]u [N ] T c kh nz ]z

Equation (48) needs only be integrated along the boundary of the computation domain where the Neumann boundary condition (24) is applied. The turbulent kinetic energy q 2 is calculated from (33) with the following substitutions: {f} 5 {q 2 },

(36)

[K] 5

E5 V

(37)

and {f} 5

Equations (32) and (33) are solved sequentially where in both cases the mass matrix [M] is obtained by (39). The potential temperature u is calculated from (32) with the following substitutions:

[K ] 5

The mass matrix [M], stiffness matrix [K], dependent variables {f }, and force vector {F} in each element can be calculated by the procedures described below. Equations (30) and (31) are solved simultaneously leading to the standard semidiscretized finite-element equation (34) in the following formats: [M] 5

E

and {F} 5

(38)

(49)

6

5 ][N ] T ][N ] 2qˆ a1 qˆlˆ 1 [N ] T [N ] dV, 3 ]z ]z B1lˆ (50)

E5 [1 V

]yˆ d ]uˆ 2 yd [N ] T 2 2u9w9 9w9 ]z ]z 1

d 2gw9 u9 uˆ

]6

dV.

2 (51)

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FIG. 1. Nonzero structure of the sparse matrix.

Here, [M], [K], and {F} in each element are integrated numerically by a 3 3 3 Gaussian quadrature approximation (Stroud 1966). They are then assembled in the global standard semidiscretized finite-element equation (34). The temporal derivative is integrated by using the Crank–Nicolson scheme (Fletcher 1991) as follows: [M]

5

6

5

6

f ti 2 f t2Dt f t 1 f t2Dt i i 1 [K] i 5 {F t2Dt }, (52) Dt 2

which can be rearranged as a system of linear equations in the form of

[ ] [ ]

M K M K 1 {f ti } 5 2 {f t2Dt } 1 {F t2Dt }. i Dt 2 Dt 2

(53)

The system of linear equations (53) is then solved by the successive-overrelaxation method (Atkinson 1989) after applying the initial and boundary conditions. Generally, by using the finite-element method discussed above, the diffusion terms of the mathematical model are solved implicitly using a centered-in-space scheme. The dissipation terms in the calculation of turbulent kinetic energy and the Coriolis terms in the momentum equations are also calculated implicitly, while the time derivatives are solved semi-implicitly. It should be emphasized that [M] and [K] are global matrices, the structures of which do not depend on the choice of [N] but the types of problems solved. In solving a one-dimensional diffusion problem (present case), the Galerkin finite-element method can be used to construct the weak formulation. It results in sparse global matrices with a bandwidth of 3, as shown in Fig. 1, where ‘‘3’’ indicates nonzero entities. The matrices [M] and [K] are symmetric except when [K] is used to calculate u and y because of the fact that [K12 ] ± [K 21 ] in (36). The large sparse matrices [M] and [K] are stored by

FIG. 2. Contour plots of the mean potential temperature u (a) observed by Clarke et al. (1971), (b) computed by Yamada and Mellor (1975), and (c) computed by the present model. The values on the contour lines (u–273) are in K.

the compressed row storage method suggested by Barrett et al. (1994) so that both the size of problem and efficiency of computation can be increased. The matrix structure is determined by the connectivity of the node points in the computational domain. This method predicts the matrix structure from the nonzero structure of the problem instead of the nonzero structure of the matrix. 5. Results and discussion a. Case 1: Comparison with the Wangara experiment The present numerical model is validated against the Wangara experiment (Clarke et al. 1971), which measured the vertical velocity and temperature profile over horizontally homogeneous terrain. Figure 2 shows the

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FIG. 3. Vertical profile of mean potential temperature u from 0900 LST day 33 to 0300 LST day 34 (a) observed by Clarke et al. (1971) and (b) computed by the present model. Here 0900 ········, 1200 , 1800 – · – · –, and 0300 LST – – –.

contour plot of potential temperature u at different altitudes and times, while Fig. 3 shows its vertical profile at different time periods. According to the results, the atmospheric boundary layer flow can be roughly divided into two types: unstable flow during the daytime (ø0800–1800 LST) and stable flow during the nighttime (ø1800–0800 LST). During the daytime, there is a very unstable layer above the ground surface, which is covered by a well-mixed layer and then a stable layer at the top. During the nighttime, a temperature inversion develops above the ground surface, which is covered by a mixed layer of nearly constant potential temperature. Comparing the potential temperature contours, it is found that the computed results of Yamada and Mellor (1975) match closer to the Wangara experiment than the present calculation (Fig. 2). The present numerical model is designed for large-scale atmospheric simulation over complex terrain; the computed vertical velocity w would be influenced by the presence of inhomogeneous terrain. However, as the present case is a one-dimensional study over flat terrain, homogeneity in the horizontal plane is assumed. Thus, in our calculation w 5 0 and the potential temperature is homogeneous horizontally. It should be noted that the assumption of w 5 0 is not always true even for flow over flat terrain. This assumption induces error in the calculation. Pielke and Mahrer (1975) showed that a vertical shear of the geostrophic wind implies a horizontal gradient of potential temperature, but its effect on the solution is not significant. Therefore, the exclusion of the vertical advection

term in the present calculation would not significantly affect the accuracy of other computed parameters. Figures 4 and 5 show the contour plots of mean wind components u (eastward) and y (northward), respectively, while their vertical velocity profiles are given in Figs. 6 and 7. According to the observations and the numerical results obtained, the atmospheric boundary layer can be divided into two layers. The first one is the surface layer found near the lower atmospheric boundary (ø0–500 m). Another region is the middle and the upper parts of the atmospheric boundary layer that are dominated by weak shear (.500 m). The surface layer is dominated by strong shear for both daytime unstable flow and nighttime stable flow. However, the wind shear in the middle and upper parts of the atmospheric boundary layer is greatly affected by its stability. Because of the strong turbulent mixing in the mixing layer, the velocity profiles are nearly constant in the mixing layer during daytime unstable flow. On the other hand, during nighttime unstable flow, the momentum fluxes in the middle and upper part of the atmospheric boundary layer are smaller than those in daytime, which leads to the formation of a local wind maximum called the nocturnal jet, as indicated in Figs. 4 and 5. After sunset, the atmospheric boundary layer cools down and becomes more stable because of the decrease in ground surface temperature. Thus, the daytime convective boundary layer gradually changes to the nighttime stable flow. The momentum fluxes decreases and a low-level nocturnal jet is developed at about 200–400 m. This type of nocturnal jet can be subsequently developed as

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FIG. 4. Contour plots of the eastward mean wind speed u (a) observed by Clarke et al. (1971), (b) computed by Yamada and Mellor (1975), and (c) computed by the present model. The values on the contour lines are in m s21.

FIG. 5. Contour plots of the northward mean wind speed y (a) observed by Clarke et al. (1971), (b) computed by Yamada and Mellor (1975), and (c) computed by the present model. The values on the contour lines are in m s21.

a consequence of free inertial oscillation (Blackadar 1957). Moreover, this kind of inertial oscillation is a simple solution to the momentum Eqs. (1) and (2) when ](2u9w9)/]z 5 ](2y 9w9)/]z → 0. As the atmosphere cools down, it becomes more stable. Both the vertical extent and the magnitude of momentum fluxes decrease, thus the nocturnal jet descends to the ground surface at midnight. Comparing the computed results with observations, it is found that the present model and that of Yamada and Mellor (1975) simulated the wind field quite well (Figs. 4 and 5). Both models can simulate the nocturnal

jet in the lower part of the atmospheric boundary layer during the nighttime, where the momentum fluxes u9w9 and y 9w9 are smaller and can be neglected. At the upper part of the atmospheric boundary layer where the momentum fluxes can be neglected, (1) and (2) changed to an inertial motion, which is dominated by the Coriolis terms. Discrepancies between the two numerical models and observations in this part of the atmospheric boundary layer may be due to the exclusion of the advection terms, which is a usual practice in homogeneous atmospheric boundary layer models. Errors may also arise in determining the geostrophic wind. Further analysis

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FIG. 6. Vertical profile of eastward mean wind speed u from 1200 LST day 33 to 0600 LST day 34 (a) observed by Clarke et al. (1971) and (b) computed by the present model. Here, 1200 , 1800 – · – · –, 0300 – – –, and 0600 LST ········.

should be conducted in order to quantify the error due to these effects. As the Wangara experiment mainly measured the mean variables u, y , and u, it is difficult to compare the calculated turbulent moments with the Wangara data. Therefore, the turbulent moments are compared only

with the results of Yamada and Mellor (1975) so as to test the performance of the model and enhance the understanding of the mean variables. Figure 8 shows the contour plots of twice the turbulent kinetic energy q 2 . The magnitude and vertical extent of q 2 increase during daytime and decrease rap-

FIG. 7. Vertical profile of northward mean wind speed y from 1200 LST day 33 to 0600 LST day 34 (a) observed by Clarke et al. (1971) and (b) computed by the present model. Symbols are the same as in Fig. 6.

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FIG. 8. Contour plots of twice the turbulent kinetic energy q 2 computed by (a) Yamada and Mellor (1975) and (b) the present model. The values on the contour lines are in m2 s22.

idly after sunset. It indicates that this kind of turbulence is associated with large convective heating or strong upward heat flux on the ground surface instead of mechanical wind shear. Another interesting feature is that the maximum value of q 2 during the daytime is located a few hundred meters above ground surface, while the maximum descends and is located near the ground surface during the nighttime. The above features can also be explained by the differential Eq. (7). The source term of (7) is constructed by a dissipation term, a buoyancy production term, and two mechanical shear terms. Assuming steady state and neglecting the diffusion term, (7) can be simplified as follows: q 3 ø B1l

[

1

2

u9w9 1 y 9w9 km

2

21

]

gw9u9 . u

(54)

The mechanical shear terms are positive so the value of the source term is controlled by the sign of the buoyancy production term. The contour plot of w9u9 shown in Fig. 9 indicates that generally w9u9 is positive during the daytime and negative during the nighttime. Thus, at night, the mechanical shear terms are counteracted by the negative buoyancy production term, while during

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FIG. 9. Contour plots of the turbulent heat flux w9u9 computed by (a) Yamada and Mellor (1975) and (b) the present model. The values on the contour lines are in units of m K s21 .

the day the positive buoyancy production terms enhance the source term. Therefore, the value and vertical extent of q 2 are greater during the daytime. In the present calculation, all the second-order moment properties are calculated diagnostically except q 2 , which is calculated prognostically. The agreement between the present results and that of Yamada and Mellor (1975) is generally good except for a greater vertical extent of q 2 obtained in the present study. Figure 10 shows the magnitudes of the temperature variance u9u9 during the daytime and nighttime. Despite the fact that u9u9 in Yamada and Mellor (1975) was obtained prognostically, while it is calculated diagnostically in the present calculation, the values obtained are comparable to each other. It shows that u9u9 can be modeled diagnostically with the use of suitable reduced algebraic expressions and hence significant savings can be made with the computation resources. Figures 11 and 12 show the wind variance components u9u9 and y 9y 9, respectively. Since q 2 is the sum of the three wind variance components, they have a similar trend as q 2 . These two wind variance components are greater during daytime, so the daytime dispersion of passive pollutant is stronger than that during

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FIG. 10. Contour plots of the potential temperature variance u9u9 computed by (a) Yamada and Mellor (1975) and (b) the present model. The values on the contour lines are in K2.

FIG. 11. Contour plots of the horizontal wind variance u9u9 computed by (a) Yamada and Mellor (1975) and (b) the present model. The values on the contour lines are in m2 s22.

nighttime. This observation is also in line with the use of greater dispersion coefficients under unstable flow (Hanna 1981).

in Fig. 13. The features of different temperature layers are similar to those mentioned in case 1 above. The present results of potential temperature are slightly higher than that of Mellor and Yamada (1974). Also, a thicker temperature inversion layer is produced above the ground surface, which is inferred as mainly due to the use of different expressions for the vertical eddy coefficient for heat. Figure 14 shows the contour plots of velocity u and y . The velocity components indicate minima and maxima in the direction of geostrophic velocity V g and U g , respectively. Moreover, the height of the nocturnal jets is about 800 m (at ø2200 LST) and decreases to about 400 m in the morning (at ø0700 LST). This type of low-level wind maxima or nocturnal jets can subsequently develop as a consequence of force inertial oscillation, as mentioned in section 5a. As can be observed from the middle and upper part of the velocity contours shown in Fig. 14, the atmospheric boundary layer thickness obtained in the present calculation is slightly greater than that in Mellor and Yamada (1974) due to the use of different vertical eddy coefficients for momentum. Nevertheless, the contour plots show a slight dif-

b. Case 2: Comparison with another numerical model The present results are compared more vigorously with the numerical results obtained by Mellor and Yamada (1974), which is a simplified case of Clarke et al. (1971). The calculation is allowed to proceed for 12 days with the ground surface temperature as the only unsteady boundary condition repeated cyclically every 24 h. For the mean and turbulent properties it requires about 10 days of calculation to obtain cyclical results. Since the ground surface temperature is the only unsteady boundary condition, the variations of mean and turbulent variables are affected only by the ground surface heat flux only. This case study is helpful in enhancing the understanding of the observations obtained in section 5a. The computed mean potential temperature u is shown

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FIG. 13. Contour plots of the mean potential temperature u for 2 days of integration computed by (a) Mellor and Yamada (1974) and (b) the present model. The values on the contour lines are in K. FIG. 12. Contour plots of the horizontal wind variance y 9y 9 computed by (a) Yamada and Mellor (1975) and (b) the present model. The values on the contour lines are in m2 s22.

ference in the velocity obtained from the two calculations. The vertical momentum fluxes u9w9, y 9w9, and w9u9 are shown in Fig. 15. The increases in vertical extent of u9w9 and y 9w9 during the daytime under convective flow can be observed. Negative values of w9u9 are observed for the stable flow, changing to positive for the unstable flow inside the convective mixing layer. This observation is in line with that obtained in the case 1 calculation. c. Comparison with water tank experiment Computed nondimensional vertical velocity variance w9w9 and heat flux w9u9 under the daytime convective boundary layer are compared with the results of Willis and Deardorff (1974) in which a convective water channel was used to model the unstable planetary boundary layer experimentally. The water channel was heated at the bottom to make it unstably stratified for the measurement of mean temperature and heat flux as well as velocity and temperature fluctuations. Figure 16 shows the computed normalized w9w9 profiles at various times (1200–1600 LST). Except the one

at 1200, these profiles have similar trends with a maximum value of approximately 0.35 at z/z i ø 0.3, which is comparable to that calculated by Deardorff (1974). Both the present model and Deardorff (1974) underpredicted the experimental value of w9w9/w*2 obtained by Willis and Deardorff (1974). This is inferred as mainly due to the use of neutral turbulent length scale l for the parameterization of the higher-order closure. Nevertheless, the computed results have the same order of magnitudes as the experimental results. Figure 17 shows strong agreement between the computed and experimental turbulent heat fluxes of Willis and Deardorff (1974). The normalized turbulent heat fluxes are negative at the top of the convective boundary layer and increased linearly with decreasing altitude. The minimum heat flux is located at the height of the convective boundary layer (i.e., z 5 z i ). 6. Conclusions A numerical model based on a second-order closure scheme has been developed and is suitable to simulate the atmospheric boundary layer flow under different stratification conditions. The model produces both mean and turbulent flow parameters under stable and unstable atmospheric boundary layer comparable to both experimental and

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FIG. 14. Contour plots of the wind speed for 2 days of integration. (a) Wind speed u (eastward) and (b) wind speed y (northward). Upper and lower figures are obtained from Mellor and Yamada (1974) and present model, respectively.

field measurement results. The model can be used to study boundary layer flow over terrain and sea breeze with several modifications such as the input of terrain elevation and surface roughness. The temperature variance calculated diagnostically in the present model is comparable to that calculated by Yamada and Mellor (1975) prognostically. This diagnostic calculation can save computer resources significantly. The present mathematical model is discretized by an isoparametric finite-element method, which is found to be very flexible in handling variable resolution meshes. This is particularly useful in obtaining local refinement of resolution in region of strong gradient. Furthermore, the use of the isoparametric finite-element method makes it easy in handling arbitrary and complex geometries and boundary. Simulation of atmospheric boundary layer over complex terrain can be easily discretized with the use of the finite-element method. The present study is focused on one-dimensional flow, which can be extended to a three-dimensional flow field for more realistic atmospheric boundary layer and air pollutant studies. Acknowledgments. The authors wish to thank the Hong Kong Research Grant Council for sponsoring this project.

APPENDIX

Nomenclature a1 , a 2 , a 3 B1 {F} 5 F i f g i iz km , kh c km , c kh [K] 5 K ij [M] 5 M ij [N] 5 N i n nz p q2 t

Constant terms in grid generation Model coefficient 527 Force vector Coriolis parameter, 2v sinc, where c is latitude and v is the earth rotation speed Gravity acceleration Vertical grid points numbering (1, 2, 3, . . . , iz ) Number of grid points in the vertical direction Vertical eddy coefficients for momentum and heat Vertical eddy coefficients for momentum and heat at t 2 Dt Stiffness matrix Mass matrix Linear isoparametric shape function 5[N1 N 2 ] Number of node points in an element The z component of the unit normal vector to the boundary G Kinematic pressure Twice the turbulent kinetic energy 5(u9u9 1 y 9y 9 1 w9w9) Time

FIG. 15. Contour plots of Reynolds stresses and turbulent heat flux. Here, (a) u9w9, (b) y 9w9 , and (c) w9u9 . Upper and lower figures are obtained from Mellor and Yamada (1974) and present model, respectively.

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FIG. 16. Vertical profile of normalized vertical velocity variance w9w9 . Willis and Deardorff (1974) case S1 – v –, Willis and Deardorff (1974) case S2 – m –, and Deardorff (1974) – – –. Computed by the present model for case 2 at various times. Here, V 5 1200, M 5 1300, n 5 1400, , 5 1500, and # 5 1600 LST.

Ug , Vg u, y , w u9w9, y 9w9 u9w90 , y 9w90 ut [W] 5 W i w9u9 w9u90 w* x, y, z z0 zi zmax , zmin

a a1 b Dt G k

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Geostrophic wind speed components in x and y directions, respectively Velocity components in x, y, and z directions, respectively Turbulent momentum fluxes Turbulent momentum fluxes on the ground surface Friction velocity Weighting function 5[W1 W 2 ] Turbulent heat flux Turbulent heat flux on the ground surface Convective velocity scale 5(z iw9u90 g /u)1/3 Space coordinates in east–west, south– north, and vertical directions, respectively Surface roughness of the ground surface Height of the convective boundary layer The z coordinates at the top and the bottom of the computation domain, respectively Angle of the surface stress vector Model coefficient 51/3 Coefficient of thermal expansion ø1/u Time incremental interval Boundary of the computation domain von Ka´rmań constant 50.35

FIG. 17. Vertical profile of normalized turbulent heat flux w9u9. Willis and Deardorff (1974) case S1 , Willis and Deardorff (1974) case S2 – – –, and the present model for case 2 at various times. Here, V 5 1200, M 5 1300, n 5 1400, , 5 1500, and # 5 1600 LST.

l l0 f fˆ {f } 5 f i V u u0

Turbulent length scale Master length scale Dependent variable (u, y , u, and q 2 ) Averaged f of each element at t 2 Dt Nodal value of the dependent variable Computation domain Potential temperature Ground surface potential temperature REFERENCES

Anderson, J. D., Jr., 1996: Mathematical properties of the fluid dynamic equations. Computational Fluid Dynamics: An Introduction, 2d ed. J. F. Wendt, Ed., Springer-Verlag, 75–84. Andre, J. C., G. De Moor, P. Lacarrere, and R. Du Vachat, 1978: Modeling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci., 35, 1861– 1883. Andreń, A., 1990: Evaluation of a turbulence closure scheme suitable for air-pollution application. J. Appl. Meteor., 29, 224–239. Atkinson, K. E., 1989: An Introduction to Numerical Analysis. 2d ed., John Wiley and Sons, 720 pp. Barrett, R., and Coauthors, 1994: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. 2d ed. SIAM. Blackadar, A. K., 1957: Boundary layer wind maxima and their significance for the growth of nocturnal inversions. Bull. Amer. Meteor. Soc., 38, 283–290. Briere, S., 1987: Energetics of daytime sea breeze circulation as de-

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termined from a two-dimensional and third-order closure model. J. Atmos. Sci., 44, 1455–1474. Businger, J. A., Y. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189. Clarke, R. H., A. J. Dyer, R. R. Brook, D. G. Reid, and A. J. Troup, 1971: The Wangara experiment: Boundary layer data. Division of Meteorological Physics Tech. Paper 19. Cotton, W., and G. J. Tripoli, 1978: Cumulus convection in shear flow—Three-dimensional numerical experiments. J. Atmos. Sci., 35, 1503–1521. Deardorff, J. W., 1972: Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci., 29, 91–115. , 1974: Three-dimensional numerical study of the heat and mean structure of a heated planetary boundary layer. Bound.-Layer Meteor., 1, 81–106. Dick, E., 1996: Introduction to finite element techniques in computational fluids dynamics. Computational Fluid Dynamics: An Introduction, 2d ed., J. F. Wendt, Ed., Springer-Verlag, 230–268. Enger, L., 1983: Numerical boundary layer modeling with application to diffusion. Part I. A two-dimensional higher-order closure model, Department of Meteorology, Rep. 70, Uppsala University, Uppsala, Sweden, 54 pp. , 1990: Simulation of dispersion in moderately complex terrain— Part A. The fluid dynamic model. Atmos. Environ., 24A (9), 2431–2446. Ferziger, J. H., 1981: Numerical Methods for Engineering Application. John Wiley and Sons, 288 pp. Fletcher, C. A. J., 1991: Computational Techniques for Fluid Dynamics. 2d ed. Vol. 1. Springer-Verlag, 401 pp. Hanna, S. R., 1981: Applications in air pollution modeling. Atmospheric Turbulence and Air Pollution Modelling, F. T. M. Nieuwstadt and H. van Dop, Eds., Reidel, 275–307. Hoffman, J. D., 1993: Numerical Methods for Engineers and Scientists. McGraw Hill, 825 pp. Koracˇin, D., and L. Enger, 1994: A numerical study of boundarylayer dynamics in a mountain valley. Part 2: Observed and sim-

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ulated characteristics of atmospheric stability and local flows. Bound.-Layer Meteor., 69, 249–283. Liu, C. H., 1998: Atmospheric boundary layer simulation and pollutant dispersion modelling. Ph.D. thesis University of Hong Kong. McNider, R. T., and R. A. Pielke, 1984: Numerical simulation of slope and mountain flows. J. Climate Appl. Meteor., 23, 1441– 1453. Mellor, G. L., 1973: Analytic prediction of the properties of stratified planetary surface layers. J. Atmos. Sci., 30, 1061–1069. , and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layer. J. Atmos. Sci., 31, 1791–1806. Moeng, C. H., 1984: A large-eddy simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci., 41, 2052– 2062. Pielke, R. A., 1974: A three-dimensional numerical model of the sea breezes over sourth Florida. Mon. Wea. Rev., 102, 115–139. , and Y. Mahrer, 1975: Representation of the heated planetary boundary layer in mesoscale models with coarse vertical resolution. J. Atmos. Sci., 32, 2288–2308. , and C. L. Martin, 1981: The derivation of terrain-following coordinate system for use in a hydrostatic model. J. Atmos. Sci., 38, 1707–1713. Sasaki, Y. K., and L. P. Chang, 1982: Methods in numerical weather prediction: Where do the finite-element methods stand? Finite Element Flow Analysis, T. Kawai, Ed., University of Tokyo Press, 27–33. Stroud, A. H., 1966: Gaussian Quadrature Formulas. Prentice-Hall, 374 pp. Sun, W. Y., and Y. Ogura, 1980: Modeling the evolution of the convective planetary boundary layer. J. Atmos. Sci., 37, 1558–1572. Willis, G. E., and J. W. Deardorff, 1974: A laboratory model for the unstable planetary boundary layer. J. Atmos. Sci., 31, 1297– 1307. Yamada, T., and G. Mellor, 1975: A simulation of the Wangara atmospheric boundary layer data. Atmos. Environ., 32, 2309–2329. Yang, X., 1991: A study of nonhydrostatic effects in idealized sea breeze systems. Bound.-Layer Meteor., 54, 183–208. Zienkiewicz, O. C., 1971: The Finite Element Method in Engineering Science. 2d ed. Vol. 1. McGraw-Hill, 648 pp.