Meteorol. Atmos. Phys, 60, 259-264 (1996) MeteorologE and Atmospheric

Physics �9 Springer-Verlag 1996 Printed in Austria

Department of Atmospheric Sciences, Nanjing University, China

Short Communication

Dynamics of the Planetary Boundary Layer with Fine Structure in the Large Scale Background Field

Y. Xu and M. Zhao

With 3 Figures

Received January 31, 1994 Revised December 12, 1995

Summary

On the basis of Wu and Blumen's work (1982) on the geostrophic momentum approximation (GMA) in the plan- etary boundary layer (PBL) and Tan and Wu (1992, 1994) on the Ekman momentum approximation (EMA) in the PBL, some improvements about the eddy exchange coefficient K, the advective inertial force and the lower boundary condition of the PBL are developed in this paper: (1) apply the K which is a gradually varying function of height instead of a constant value in the Ekamn layer, and introduce a surface layer; (2) take the effect of the vertical advective inertial force into account; (3) the solution technique is extended from level terrain to orographically formed terrain. Under the condi- tion of the equilibrium among four forces (the pressure-- gradient force, Coriolis force, eddy viscous force and inertial force including horizontal and vertical advective inertial forces), we have obtained the analytical solutions of the distributions of the wind and the vertical velocity. The com- putation of an individual example shows that: (1) both the wind velocity near surface and the angle between which and the non-viscous wind are more consistent with usual obser- vations than that of Wu and Blumen (1982); (2) comparing with the horizontal advective inertial force, the vertical ad- vective inertial force can not be neglected, when the orogra- phy is considered, the effect of the latter is even more important than the former.

1. Introduction

In the traditional theory of the PBL, the effect of the inertial force is usually neglected in the re-

search of dynamics of the PBL for a long time. Wu and Blumen (1982, 1991) and Panchev et al. (1987) introduced the G M A into the PBL. The effect of the advective inertial force on the distribution of the wind within the PBL can be expressed ex- plicitly, promoting the developments of the dy- namics of the PBL and its application. On the basis of the GMA, Tan and Wu (1992, 1994) introduced a very convenient method named the Ekman momentum approximation, but the verti- cal advective inertial force has been neglected in their works. However, in the PBL, the horizontal and vertical advective inertial forces are of the same order. Therefore, including the vertical ad- vective inertial force can further improve the dy- namics of the PBL. The distribution of the wind in the PBL is very sensitive to the eddy exchange coefficient K. In the modern PBL model, the distribution of K can be expressed by different formulas. For example, K can be expressed through a cubic function of the height z (O' Brien, 1970) or a quadratic function (Nieuwstadt, 1983). Grisogono (1995) introduced three non- rapidly-varying K-profiles, one of which corre- sponds to O' Brien's (1970). In this paper, the PBL is divided into two sublayers: surface layer and Ekman layer. In the Ekman layer, K is taken as a quadratic function of z, as an assump-

260 Y. Xu and M. Zhao

tion:

= ( . z + (1)

where c~,/~ are determined by the geostrophic wind speed (see below). By using this K, it is very convenient to get the analytical expression of the distribution of the wind in the Ekman layer with a primary complex function of K. This treatment can not only simplify the calculation, but also analyze the dynamical process conveniently. Based on this improved EMA, we give a simple example to analyze the distributions of the wind and the vertical velocity in a barotropic and neu- tral PBL with and without orography. Some pri- mary conclusions which are roughly in agreement with usual observations are obtained.

2. The Wind in the Surface Layer and the Modified Ekman Wind

For simplicity, we discuss in a barotropic and neutral PBL. Dividing the PBL into the surface layer and the Ekman layer is often adopted in the boundary layer model (Estoque, 1973), their tops are marked as h 1 and hr respectively, in general, they can be taken as 100 m and 1000 m respective- ly. In the neutral PBL, usually the eddy exchange coefficient K within the surface layer is taken as the linear function of z and the wind distribution there satisfies the logarithmic law.

Following the classical Ekman theory, within the Ekman layer, the wind satisfies the equilib- rium among three forces: the pressure - gradient force, Coriolis force and eddy viscous force. We take K as the quardratic function of z which is more appropriate compared with the constant K, that is Eq. (1). Supposing that K has its maximum value K,, at h~, and taking the value of K at h r as follows:

K r = K(hr) = mKm, (2)

where m is a constant between 0 and 1, we obtain:

= K~m/2(m 1/2 - 1), fl _ K~/Z(hr - h l m 1/2) (3)

h r - h I h r - h 1

Using the continuities of the wind and the eddy stress at the level z = h 1 as the connecting condi- tions, we can get the wind velocity solutions W 1 within the surface layer and W e within the Ekman

layer as follows:

W1 = W . In z , (4a) g Z o

W~ = W o + Wo(aK r/2 + a'Kr'/2), (4b)

where

a = [eH(r - r'mU)K~/2 + m ~ - l-I - 1K~,~/2, a'

= - ( m K , , ) " a ,

W . = ~ceh t Wo(r - r' mU)K~- 1)/2a,

1 h a H = i n - , (5)

z o

1 1 1 [ . 4 f ~ 1/2 r = - ~ + # , r ' - ~1+ 2 #' #=-2 l -d~) "

The value of K x corresponds to that of K at z = l m . W. is the complex friction velocity, ~c the Karman constant, z 0 the roughness. W 0 is the complex geostrophic wind, W o = u o + ivo, uo, v o are the horizontal velocity components in x and y directions respectively. W e is the wind in case of the equilibrium among three forces and with a varying K (Eq. (1)) within the Ekman layer which can be called modified Ekman wind or the wind under the equilibrium among three forces. The continuity equation in the PBL is

~w c~u + ~-Y , (6) v-W+ z =0, v.w=

substituting the wind solved into Eq. (6) and integrating from z o to arbitrary height z within the Ekman layer, we can obtain the modified Ekman vertical velocity we(z) as

We(Z ) - D * ( h l l n h - ! - h , z o

- [uoV 'B ( z ) + voV'( iB(z)) ] + [oBi(z), (7)

(B(z) = Br(z) + iBi(z )

l [ r ~ l (K(~+ 1)/2 ) K~ +1,/2 )

+ ~ ( K ( ~ ' + 1)/2 _ K~'+ 1)/2) , r t l

D, = V" W,).

3. The Wind under the Equilibrium among four Forces

In the PBL, the motion under the equilibrium among four forces (pressure-gradient force,

Dynamics of the Planetary Boundary Layer with Fine Structure 261

Coriolis force, eddy viscous force and inertial force) is

w . v + w =

w+ fK w) i f W 0 - i f ~z\ ~z]" (8)

The method we deal with this non-linear problem is similar to Tan and Wu (1992, 1994) who sugges- ted the EMA instead of the GMA to treat the PBL equation. In the free atmosphere, the geostrophic wind W o represents the basic state of the atmo- spheric motion, the advected velocity W in the motion equation can be replaced by Wg approxi- mately in the GMA. However, in the PBL, the basic motion is not geostrophic but Ekman flow, thus the wind advected W in Eq. (8) should be replaced by W E not Wg, and Win the eddy viscous term can be replaced by WE (Tan and Wu, 1994), so we can get the linearized motion equation asi

~ --~- w " g -[- We-~z

i f W ~ -~-z ] ' (9)

here we have replaced the vertical velocity w with w e approximately to reduce an unknown variable w. Because WE is the wind under the equilib- rium among three forces, so Eq. (9) can be re- written as

( 0 0 ) W e = i f WE-i f W , (10) ~ nt- w " g -Jf- We ~ Z

o r

Au + By = C, (11)

where A, B, C are complex constants as follows:

A = a I + ia2, B = b 1 + iba, C = C 1 + i c2 ,

Oue OUe bl = f , a l = - 0 ~ ' Oy

~b/e ~Ue (12) C l -'= f V e "+- - ~ --q-we O Z '

~Ve b2 = ~Ve a 2 = - f 8 x ' - O'--fi'

~V e ~1) e C 2 = -- fUe + ~ - + w e OZ'

Eq. (11) can be easily solved and the result is noted as W2. So far we have got the solutions W~ (in the surface layer) and W 2 (in the Ekman layer) as follows:

w** 7 ;

W1 - In ---, (13a) K Z o

C-(iB) C.(iA) (13b) W2-A.(iB) i A-(iB)'

where

K W * * - l n ( h l / z o ) W2(hl). (14)

4. The Wind and Vertical Velocity in the PBL over Orography

For the PBL over orography, Wu (1985) postu- lated that the wind velocity at the mountain slope W~/was roughly equal to a small fraction of its value at the same level when the mountain was absent and which was taken as the lower bound- ary condition as follows:

Z = z o + h g , W I = W e ` = y W o, W h = W e , V h g , (15)

where Z is the height above sea level, h o the profile function of orography, V a small fraction between 0 and 1 which is empirically determined. W o is the wind velocity in the PBL over the flat surface, w h

the vertical velocity at the slope surface of the mountain caused by the forced lifting of oro- graphy.

Taking the following coordinate transtbrma- tion;

z = Z -- ho, (16)

we can easily obtain the solutions in the PBL over orography under the equilibrium among three forces as

W~ = W n + W , In z , (17a) h: Z o

W e = W o + ( W o - We` ) (aK r'/2 + a'Kr'/2). (17b)

where a, a', W. are the same as in Eq. (5), but W 0 in Eq. (5) should be replaced by (W 0 - We`).

Integrating the continuity equation and using Eq. (17), we can find the modified Ekman vertical

262 Y. Xu and M. Zhao

velocity w e as

We(Z ) : Dh(Br(z ) - h I + z O)

~ * ( h l In z ~ - h i + z 0 )

- - [ (u o - uh) V . B ( z ) + (v o -- v h ) V ' ( i B ( z ) ) ]

+ ((0 - ~h)B,(z) + W E ' V h o ,

(D,, = V" W n ) . (18)

The solution in the PBL over orography under the equilibrium among four forces are:

W 1 = W. + W** in z , K Z o

(19) C-(iB) C'(iA)

W2 - A.(iB) i A.(iB~'

where A, B, C are the same as Eq. (12), but the (u e, v e, %) in Eq. (12) should be replaced by Eq. (17b) and Eq. (18).

By integrating the continuity equation, the ver- tical velocities w 1 in the surface layer and w 2 in the Ekman layer may be written as: o**(z ) w 1 = Dh(z -- zo) -- - - 7 - z In - - -- z + z o

Z o

+ W I . V h o,

zo,% In +4

;5 - V" W z d z ~- W 2 " V h o, 1

(D** = V- W**). (20)

Up to now, we have got the wind and the vertical velocity in the PBL over orography under the equilibrium among four forces, as to the deter- mination of the value of K m, we may use the implicit equation about Km as follows:

K m = ~chl I W** I. (21)

,It is easy to get Km by iterative method, thus K,, is concerned with the pressure field. This shows that the eddy exchange coefficient K in this model is not only the function of z, but also t, x, y, i.e., the large scale background field.

5. Calculation and Analysis - an Example

For simplicity, a steady, neutral and barotropic atmosphere is considered. The geostrophic wind

can be taken as

ug = constant,

27z v o = A d cos ~ x.

(22)

Set the wave length L = 3 x 10 6 m, the roughness z o = 0.1 m, Coriolis parameter f = 0.0001/s, the constant factor in Eq. (2) m = 0.001, ug = 15m/s, A d = 6 m/s.

We calculate at x = L / 4 and 3 L / 4 which corre- spond to negative and positive geostrophic vorti- city ((0= -T 1.25 x 10-5s -1) areas respectively. For these two points, (u o, v0) are the same, so the wind spirals there under the equilibrium among three forces are the same too. The wind spirals under the equilibrium among three and four for- ces are shown in Fig. 1, curve a is the wind spiral under the equilibrium among three forces (by Eq. (4)), curves b, c among four forces at x = L / 4 and 3 L / 4 respectively (by Eq. (13)). For comparing, the classical Ekman spiral curve ao is depicted in Fig. 1 (K = constant = 10 m2/s, z o = h 1 = 0, h r ~ oo).

In Figs. 1 and 3, the direction of the abscissa axis is taken to be parallel to the geostrophic wind at the top of the PBL, and the numbers at curves denote the height in meter.

The distributions of the vertical velocities under the equilibrium among three and four forces with and without orography are shown in Fig. 2. For simplicity, we take a slope of Ohg/~?x = 0.001 and set the wind at the slope to be zero.

Curves a, a' depict the distributions of the vertical velocities under the equilibrium among three forces by Eq. (7) without orography and Eq. (18) with orography respectively, curve b' among four forces with orography by Eq. (20) at x = L / 4 , curve b by Eq. (20) with h o =- O.

Figure 3 is a diagram of the wind spiral with orography under the equilibrium among four for- ces at x = L/4 . Curves b' and b" are obtained by Eq. (19) but curve b" is for ignoring the part of the vertical advective inertial force. Figure 3 shows the importance of the vertical advective term in case of orography.

6. Conclusion

In this paper, under the consideration that the eddy exchange coefficient K varies with height and is connected with wind, the analytic solutions

Dynamics of the Planetary Boundary Layer with Fine Structure 263

400 400 5 - . . . . . . . . . . . ~'~,C'" ~;'-*"~".'=..; . . . . . ~ ( ' " . . , . .

" ~ 200 . . . . . . . . . . . . . 20_0 ._..~..~ X '~" - - - " ' - -o . : -1-~= " ' " - . . . . ..z:"" ,oo ~oo ~ . . . . ?" - - - , , . - "~ .~ . " ' - . . 600

�9 . . " ,.o "'... , , . ,

4 - , ..... 50 ";~ ,~ , ,o - ,oo ~oo "-.. ~oo z , - - . . 2 o o - ~ .

�9 ~" 5 '~v_~.-"~" 20 \ 7B0 ~ . \ .."" ~ _: ._r ,o \ ~ ? ~ . . \ \

2 - . . . ' " ^ 2 ~ , ~ ' 5 -~,~ ~ 9an ZX 760 50,." 1 Z ~ . ~ ~,~u �9 ",' ; o

,ooo . \ i - N ix 2 ~ . . o V , ; i I' 82o

z" '?~"" , .: t �9 " ~ ' I I ~ I I i I I i I I ,i I '~176176 ] I ooo t j ,9ool

O r 1 2 3 4 5 6 7 8 9 10 11 12 ~,13 14 .J15,o...16 17 '~e ~"~00 960

u(m/s) 96o

"t760 X

x 820

i I ,u

18

Fig. 1. The wind spirals under the equi l ibr ium a m o n g three and four forces (curve a, three forces; curve b, c, four forces at x = L/4 and 3L/4 respectively; curve a0, classical E k m a n spiral, K = constant , h~ = z 0 = 0, h r ~ m)

1000

900

800

E 700 v I'4

600

5O0

400

3O0

2O0

I00

0

/?

/ r7

I 1

i [ J , 1 2 3 4 5

~~ f"

I~.X" I

f " .f*

oX"

~ g ~ " l ~ ~ t " [ " I L i I { I 1 I 1 6 7 8 9 10 11 12 13 14 15 16 17 18

lw I (10-3m/s)

Fig. 2. T h e d i s t r i bu t ion of the vert ical velocit ies wi th height . (curves a, b w i t h o u t o r o g r a p h y , curves a', b' wi th o rog raphy ; a, a' the

equ i l ib r ium a m o n g th ree forces; b, b' a m o n g four forces a n d x = L/4)

of the wind in the PBL under the equilibrium among three and four forces are found. After finding the solutions under the equilibrium among three forces, the solution under the equilibrium among four forces is obtained approximately by an algebraic equation and the corresponding wind spiral is more reasonable

physically than that of Wu and Blumen (1982), the solving method is very simple which has certain theoretical and applied meaning. The calculated results prove that either for the wind spiral in the PBL or the vertical velocity, the vertical advective term involved in the inertial force terms cannot be ignored compared with the horizontal advective

264 Y. Xu and M. Zhao: Dynamics of the Planetary Boundary Layer with Fine Structure

G

400

6 , --" ~," ' ~ . 600 100 -~ " " ' ~ It"

~,~'" 400 "'~ 50 ..-" 200 ~ . . . . . . . . . . . , ~ . . ~ . " " .

~ " 200 , ,X " ' ~ ' " 400 " ~ . . ~ . \ 20 /./" lOO~od~'~"'-'--"-~ "~"... 600 "% b ~

- Io 1 -~c so/x'~ ~ 600 "x . . . . "\ 5 t Jr" 20 _w,"'x'~l O0 " o ~ . " 'N b " \

.~s" zo J~o ~ '\. "\ 76 z , , . " s . . - x ' ~ o N . -~ o ~t' 2 o . , .X" lO ~ : 76{)

I . /" / ~ ~ 7 6 0 x.. .

2s --'~ ~'s" , 2 ] ,~ 820 ! 820 1 u. ~.. oX"" l o 820 : ,~

~'~' 25,0. 25 1, J [ 1 ] ] [ [ J ] I ] 1000 ,lxOO0 J / ( . ] ] / ] =__

1 2 3 4 5 6 7 8 9 I0 Ii 12 13 14 15...ia 17". 18/g19 20 21:" 22

960 960 . / u(m/s) /

If" 960

Fig. 3. The wind spirals with orography under the equilibrium among four forces at x = L/4, but curve b" is for ignoring the vertical advective inertial force

term, furthermore, which takes a more important role in the inertial force terms when the orography is considered; our results improve the results of Wu and Blumen (1982), and Tan and Wu (1994).

R e f e r e n c e s

Estoque, M. A., 1973: Numerical modelling of the PBL. In: Workshop on MicrometeroIogy. Amer. Meteorol. Soc., 217-270.

Grisogono, B., 1995: A generalized Ekman layer profile with gradually varying eddy diffusivities. Quart. J. Roy. Meteor. Soc., 121, 445-453.

Hoskins, B. J., 1975: The geostrophic momentum approxi- mation and the semi-geostrophic equation. J. Atmos. Sci., 32, 233-242.

Nieuwstadt, F. T. M., 1983: On the solution of the stationary baroclinic Ekman layer equation with a finite boundary layer height. Bound:Layer Meteor., 26, 377-390.

O'Brien, J., 1970: A note on the vertical structure of the eddy exchange coefficient in the PBL. J. Atmos. Sci., 27, 1213- 1215.

Panchev, S., Spassova, T. S., 1987: A barotropic model of the Ekman planetary boundary layer based on the geo-

strophic momentum approximation. Bound.-Layer Me- teor., 40, 339-347.

Pedlosky, J., 1979: Geophysical Fluid Dynamics. New York: Springer.

Tan Zhemin, Wu Rongsheng, 1992: Dynamics of Ekman momentum flow and frontogenesis. Science in China, Seri B, Vol. 35, 115-128.

Tan, Z., Wu, R., 1994: The Ekman momentum approxi- mation and its application. Bound.-Layer Meteor., 68, 193-199.

Wu, R., Blumen, W., 1982: An analysis of Ekman boundary layer dynamics incorporating the geostrophic momentum approximation. J. Atmos. Sci., 39, 1774-1782.

Wu, R., Blumen, W., 1991: The Ekman boundary layer over orography: analysis of vertical motion. Bound.-Layer Me- teor, 54, 315-326.

Wu, R., 1985: The influence of orography upon the flow within Ekman boundary layer under the approximation of geostrophic momentum. Adv. Atmos. Sci., 2, 1-7.

Authors' address: Xu Yinzi and Zhao Ming, Department of Atmospheric Sciences, Nanjing University, Nanjing 210008, China.

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