Notes on the Turbulence Closure Model for Atmospheric Boundary Layers

Transcription

1 466 Journal of the Meteorological Society of Japan Vol. 56, No. 5 Notes on the Turbulence Closure Model for Atmospheric Boundary Layers By Kanzaburo Gambo Geophysical Institute, Tokyo University (Manuscript received 24 March 1978, in revised form 12 July 1978) Abstract A turbulence closure model for atmospheric boundary layers is examined, under the assumption that the turbulent flow is steady in its ensemble average and the advection and diffusion terms in the turbulent Reynolds stress and heat flux equations are neglected. The constants which are introduced in order to obtain a closure system are determined referring the experimental results obtained in the wind tunnel. The validity of the closure model thus obtained is checked referring the observational results obtained in the constant-flux layer. In order to apply our model to the planetary boundary layer, simple forms for the eddy transport coefficients of momentum and heat are formulated. In this case the parametarization of the scale-length l=l(z) (z: heihgt) which is introduced in estimating the dissipation rate of turbulent kinetic energy with height is discussed in order to satisfy the observational result. 1. Introduction Recently the so-called closure models which use exact equations for the mean field and approximate ones for the turbulence have been successfully applied to predict the detailed structure of the planetary boundary layer (for example, Mellor and Yamada (1974), Yamada and Mellor (1975), Wyngaad, Cote and Rao (1974), Deardorff (1974a, 1974b), Wyngaad (1975), Schemm and Lipps (1976), Sommeria (1976)). In the closure model, however, there is the choice how to select the empirical constants which are used to represent the approximate equations for the Reynolds stress and heat flux due to the turbulence. We examine the choice of the empirical constants in the closure model in this paper. In this note we proceed our discussion refering Mellor's (1973) paper, because he is the first to apply the closure models which were developed in the field of engineering to the prediction of the properties of the constant flux layer in the atmosphere. Mellor determined the empirical constants from the data under the neutral stability condition. Generally speaking, the estimation of the pressure-strain correlation in the Reynolds stress (uti uj) equation and the pressure-temperature correlation in the heat-flux (ui*) equation is essential in the success of the closure model. Therefore, the pressure-strain correlation in the Reynolds stress equation is re-examined in the first part of this paper, comparing the treatment proposed by Mellor. After that the pressuretemperature correlation is also re-examined. In Mellor's case, it is assumed that there is no direct influence of gravity on the pressure-containing correlation in the equations for the Reynolds stress and heat-flux. The present note argues that such an influence should be included, refering Launder's (1975) proposal. In the latter part of this paper, our model is extended to predict the properties of the planetary boundary layer and we present the parametarized scheme for the eddy coefficients of momentum and heat, KM and KH, introducing the scale-length l. In the constant flux layer, the scale-length l is l*kz (k: Karman's constant) outside the roughness sublayer. In the planetary boundary layer, however, the scale-length l is l(z). Therefore the stipulation of the functional form of l(z) should be considered in order to satisfy the observational data such as w2 (vertical velocity variance) and *2 (potential temperature variance).

2 October 1978 Kanzaburo Gambo The basic equations The equations of motion for the mean velocity and mean potential temperature may be written i n the form they are small in the treatment of planetary boundary layer. On the other hand, the equation of *2 may be written in the form U* and u* are mean and fluctuating velocity component (i, j, k run from 1 to 3, *1*, 2*y and *3*z in (x, y, z)-coordinate * system), and * are mean and fluctuating potential * temperature and P is the mean pressure. The overbars represent ensemble averages. The Boussinesq approximation and the adiabatic motion have been made here, and *0 and. *0 are constant representative values of density and potential temperature respectively. *ij* is the alternating unit tensor, fi= (0, 0, f) is the Coriolis parameter, g is the gravitational acceleration, and * is the Kronecker delta. Molecular diffusive terms have been neglected. The equations govering the transport of the Reynolds stress u* uj and heat flux ui * may be written in the form 3. Approximation of the pressure-strain correlations. The equations govering the Reynolds stress and heat flux contain unknown correlations of fluctuating quantities such as shown in (2.3), (2.4) and (2.5). These must be approximated in terms of the main dependent variables if we wish to obtain the closed system for the govering equations (2.1)*(2.5). In this section we consider the case there is no buoyant force for the sake of simplicity and discuss the parameterization of the pressure-strain correlation in (2.3), Concerning this problem many proposals have been discussed and each proposal seems to be based on the different assumption. Since the detailed review of individual proposal is not main problem of this paper, we omitt the reference of these proposals. However, it might be generally accepted that the principle of all proposals is based on the parameterization of the pressure-strain correlation proposed by Rotta (1951). Therefore we pick up here Rotta's original proposal in order to discuss the closed system for (2.3) systematically. For the estimation of pressure-strain correlation, Rotta proposed the following form for (p/*0)(*ui/*j) in the homogeneous turbulence: p is the pressure fluctuation about mean value, is the kinematic viscosity and * is the kinematic * thermal diffusivity. The terms which contain the Coriolis term have been neglected here because Here C1 is a constant and E and * are the timeaveraged turbulent kinetic energy and energy dissipation rate respectively, i.e.,

3 468 Journal of the Meteorological Society of Japan Vol. 56, No. 5 in a case of isotropic turbulence in (2.3). The quotient E/* thus represents a characteristic decay time of the turbulence. The second term on the right hand side of (3.1), *ij2 arises from the presence of mean rate of strain and the coefficient aljmi is approximated by determined to satisfy the experimental results of the nearly homogeneous shear flow obtained by Champagne et al. (1970), Ui= {U, 0, 0} and U varies linearly with *3 (*3*z). And the most reasonable constants are given by the *'s are the Cartesian components of the position vector X-Y and term with and without a prime relate to values at Y and X respectively (the integration being carried out over Y space). Rotta pointed out that the fourthorder tensor {aljmi} should satisfy the following kinematic constrains: In a special case Ui*{U, 0, 0}, U= U(*3), and the homogeneous turbulence is assumed, Rotta computed approximately the values of a1131, a1232, * as follows: provided that C1 takes a value of about 1.5*. Considering that C2*0.4 in (3.9), it is easily understood that the first term on the right-hand side of (3.9) predominates compared with other terms. Then Launder et al. proposed the approximate expression for (3.9) as follows: * is a constant. It is expected that the constant * will differ somewhat in magnitude from the coefficient of the first term in (3.9) (*8.4/11 for C2*0.4) to compensate in part for the neglected terms. In a case Ui= {U, 0, 0} and U=U(*3), we have P=-u1u3(*U/*3) in (3.12) and then Later Hanjalic and Launder (1972), and Launder, Reece and Rodi (1975) extended Rotta's treatment in a general form. Since the forms of (3.5)*(3.7) suggest that aljmi might be satisfactorily approximated by a linear combination of Reynolds stress, Launder et al. proposed the following form for *ij2 in the homogeneous shear flow: Comparing these values with those in (3.8) obtained by Rotta, the appropriate value of * will be given by 0.6 * In this case (*11)2, (*22)2 and (*33)2 give the same result as Rotta's estimation. However there is a little difference in the estimation of [(*13)2+ (*31)2], because Rotta's estimation is given by and P denotes the rate of production of turbulence energy. C2 in (3.9) is considered to be a constant. The constants C1 and C2 in (3.2) and (3.9) were Considering the discussion mentioned above, we use the equation (3.12) in the present paper for estimation of the pressure-strain correlations. Now we consider to determine * from the another point of view. If we neglect the transport and buoyancy terms in (2.3), incorporate the pressure-strain approximation such as shown in (3.12) and the dissipation term is expressed by 2/3(*ij), the equation (2.3) reduces to a set of following equations for the normal stresses: * Rotta originally proposed that C1 should be about 1.4.

4 October 1978 Kanzaburo Gambo 469 may be written in the form Here we use the relation of *=-u1u3(*u1/*3) and the values denoted by parentheses of the last terms in the above equations give the experimental result of Champagne et al. Under the assumption of C1=1.5 given by Launder et al., we have from the above relations between ui2 and * Thus the pressure fluctuation due to the effect of temperature fluctuations is expected in the case of buoyant flow and Launder proposed the pressure-strain interaction as follows: Considering the above estimation of *, we use * =0.65 and C1=1.5 tentatively in this paper. As pointed by Launder et al., the equation (3.9) is reduced, irrespective of the value of C2, to the following equation: provided that uiuj=2/3 (E*ij). This equation is the same as that obtained by Crow (1968) for the case of isotropic turbulence subjected to sudden distortion. If *=0.6 and uiuj=2/3(e*ij), the equation (3.12) is also reduced to the same equation as (3.15). In this sense, the expression such as (3.15) seems to be a special form for the estimation of (*ij2+*ji2). Here we note that the equation similar to (3.15) is usually used in the treatment of turbulence in the planetary boundary layer (Mellor, Mellor and Yamada, Deardorff (1975), Som- (1974a), Wyngaad et al., Yamada meria, Schemm and Lipps). 4. Analysis in buoyant shear flows In the preceeding section we discussed the approximate expression of the pressure-strain interaction for the case of non-buoyant flows. In this section we consider the extention of the above discussion to the case of buoyant flow. The main part of this section is due to Launder's proposal and some part of his paper is referred here, because we are particularly concerned with the process how the empirical constants are reasonably introduced in the parameterization of the pressure-strain and pressure-temperature correlations in (2.3) and (2.4). The divergence of the momentum equation for ui under the assumption of incompressible fluid Here Pij and P are considered to stand for the total generation of uiuj and E due to the combined effects of shear and buoyancy. The value of C2 is put as C2=*. Consistently, the same consideration is adopted in approximating the pressure scrambling term, (p/*0)(*/ *i) in (2.4). And we put C1T and C2T are constants, and In the usual closure mode, the second term of the right-hand side in (4.5) is neglected, i.e., C2T =0. However, we assume that C2T*0 in parallel with (4.2). Next we consider the dissipation term in (2.3), (2.4) and (2.5). Assuming isotropy of the fine scale motion, the dissipation term in each equation may be written CT' is a proportional constant. In esti-

5 470 Journal of the Meteorological Society of Japan Vol. 56, No. 5 mating CT', Launder used the following relation: which is derived from (2.5) and (4.9) in equilibrium flows, neglecting the diffusion term in (2.5). From the decay characteristics of ui2(=2e) and * 2 behind the grid, CT' is estimated by Launder as and CT'*1.6 (4.11) Here we note some comment about CT'. We have from (4.7) and (4.9) The hydrostatic relation is used in (5.3). Now we consider to rewrite the equations (2.3), (2.4) and (2.5), referring the analysis in section 4. In section 4, we use E/ * as the parameter which represents a characteristic decay time of Hinze (1975) estimated as CT'=4/3*1.33 under turbulence. In the so-called closure model, the the assumption of the "linear" velocity decay. prognostic equations for * is usually used to close On the other hand, Mellor assumed that CT'= the system which we use. In this paper, however, 16/15*1.07*, referring the boundary layer measurements of Johnson (1956). In this sense, there we assume the following simple relation between and E, * remains uncertain factor in determing the value of CT'.** In this paper, however, we use tentatively the value of CT' such as CT'=1.6. l is the characteristic scale length and C Concerning the estimation of C1T and C2T in is a constant. Then we have (4.5) Launder used the following relations Comparing these relations with Webster's (1964) experiment (for example, 1/CT'C1T*0.2), Launder estimated as C1T*3.2, C2T*0.5 (4.15) 5. Govering equations in the boundary layer If we apply the discussion mentioned in sections 2, 3 and 4 to a boundary layer the vertical scale is much less than the horizontal scale, equations (2.1) and (2.2) may be written as follows: and the equations (2.3), (2.4) and (2.5) may be written in the form * Following Mellor's notation we have CT'/2= B2/B1=8/15 in (4.12). ** The value of CT' is sensitive for the estimation

6 October 1978 Kanzaburo Gambo The constant flux layer If the mean velocity field is assumed as Ui= {U, 0, 0) and U=U(z), the Reynolds stresses and heat fluxes in the constant flux layer may be put in the following way: As mentioned in sections 3 and 4, the parameters C1, C2, CT', C1T and C2T are given in this paper as follows: In the constant flux layer, we assume further that the diffusion terms in (5.8), (5.9) and (5.10) are neglected. Then the equations for u2=u12, *2= u22, w2=u32 and uw=u1u3 in (5.8), u*=u1* and w*=u3* in (5.9) and *2 in (5.10) may be rewritten in the nondimensional form as follows: In order to fix the value of C in (5.8), (5.9) and (5.10), we use the equation (5.8), assuming that /*t=0, Ui= {U, 0, 0}, C1T=C2T=0 (no bouyant * flow) and the diffusion term is neglected. Since we assume no buoyance, we have From the equation for u1u3(=uw) in (5.8) we have On the other hand we have from (3.14) Elimination of w2 and E from (5.14)*(5.16) gives the following relation between uw and U/*z: * If l=kz (k: Karman's constant) and G(C)=1, the equation (5.17) corresponds to the formula, given by Prandtl's mixing length theory. Thus the value of C is given by G(C)=1, i.e.,

7 472 Journal of the Meteorological Society of Japan Vol. 56, No. 5 As mentioned in section 5 (cf. (5.17) and (5.19)), the value of C is determined under the assumption that l is proportional to z outside the roughness sublayer and we put = kz l (6.18) k is the Karman's constant. Now we consider to obtain the value of *M and *H as the function of *, eliminating u*2, *2 *, w*2, u* **, **2 from (6.4)*(6.10). By making the addition of (6.4), (6.5) and (6.6), we have the energy equation such as M =*+CE*3/2 * (6.19) Therefore we have in a strongly unstable case (-*>>1) provided that C=0.203 as mentioned in section 5. If we eliminate **, w* and *M from (6.6), (6.9), (6.10) and (6.19), we have Fig. 1 Calculated values of *M (solid curve). Dotted curve is obtained by Mellor. The equation for *M is obtained by eliminating u*** and w* from (6.6), (6.7) and (6.8) as follows: Eliminating E* from (6.19), (6.21) and (6.23), we obtain *M(*) and *H(*) such as shown in * The value of CT' is very sensitive for the estimation of *H on the stable side. A variation, CT'/CT'=0.2 products a variation *H/*H*0.3. * Fig. 2 Calculated values of *H (solid curve). Dotted curve is obtained by Mellor. Figs. 1 and 2.* In order to compare our results with those obtained by Mellor we show his results by dotted line in Figs. 1 and 2. In Mellor's case the type of (3.15) is used to estimate the pressurestrain correlation and the assumption that C2T= 0 in (4.5) is used. In Figs. 3 and 4, the values of *M and *H on the unstable side (*<0) are shown by the contineous line, while Mellor's case corresponds to the dotted line. For the sake of comparison, the empirical formulas such as *M4-9*M3=1 (KEYPS) and *M=(1-15*)-1/4 (Bussinger et al. (1971)) are shown by dot-chain lines in Fig. 3. The observed data plotted in Fig. 4 are quoted from the paper of Bussinger et al.

8 October 1978 Kanzaburo Gambo 473 Fig. 3 Comparison of the calculated values of *M (solid curve) with interpolation formulas (dot-chain curve) under unstable conditions (*<0). Dotted curve is obtained by Mellor. Fig. 5 Comparison of vertical velocity variance *W*2 with data assembled] by Wyngaad et al. (1971). Dotted line is obtained by Mellor. Fig. 4 Comparison of the calculated values of *H (solid curve) with interpolation formula (dot-chain curve) and observed data (from Businger et al., 1971) under unstable conditions (*<0). Dotted curve is obtained by Mellor. (1971a) and the dot-chain line in Fig. 4 corresponds to the empirical formula *H=0.74 (1-9*)-1/2 (Businger et al.). The values of *w*z and **2 are also shown in Figs. 5 and 6, refering the Mellor's case by dotted line. The observed data in these figures are quoted from the paper of Wyngaad et al. (1971). As may be seen in Figs. 3*6, our results show the good agreement with the observational data in unstable conditions (*<0). It may be suggested that the treatment of the pressure-strain correlation such as (4.2) plays an important role to obtain the reasonal values of *H(*) and *M(*) compared with those obtained by Mellor. In stable conditions (*>0), there is discrepancy about the values of *H and *M between Mellor's result and ours. Concerning the value of *H we Fig. 6 Comparison of temperature variance with data assembled by Wyngaad **2 et al. (1971). Dotted line is obtained by Mellor. notice that *H is sensitive with respect to CT' in stable conditions and *M is not sensitive with the value of CT'. As mentioned in section 4 there is uncertain factor in determining the value of CT'. If we put CT'=1.2 instead of CT'=1.6, the value of *H on the stable side in Fig. 2 gives the almost same value as that shown by dotted line in the figure. Concerning the discrepancy of *M in Mellor's case and ours, we must mention that there is no unique empirical formula of *M in stable conditions (*>0) to compare the theoretical result. In the discussion of *M and *H mentioned above, we have only refered the empirical formulas proposed by Businger et al, as one

9 474 Journal of the Meteorological Society of Japan Vol. 56, No. 5 example. Concerning the empirical formulas of M and *H, however, Pruit et al. (1973) and * Kondo (1962, 1975) proposed the following ones: (b) stable case: Generally speaking, these empirical formulas including that of Businger et al. show the difference in considerable magnitude in stable conditions (*>0), though the difference is small in the unstable conditions (*<0). For example, we have for *=1 M*5.7, * *H*5.44 Businger et al. M*2.2, *HN*3.0 Pruit et al. * Yamamoto and Shimanuki (1966) and Yamamoto (1975) proposed the generalization of the KEYPS formula in order to explain the discrepancy mentioned above. In these circumstances, we do not discuss here in detail about the difference between the empirical formula and the theoretical result in stable conditions. Concerning the value of **2 on the stable side in Fig. 6, both Mellor's case and ours show that **2 increases with *, while the observation shows the slowly decrease of **2 with *. This comes from the fact that **2 is proportional to H/E*1/2 such as shown in (6.10) and the increase of *H is larger than that of E*1/2 with * respect to * (>0). Anyway the problem to be revised still remains in stable conditions. In order to understand the characteristic feature of *M and *H we compute the values of *M and *H when -*>>1 and *=0. In case -*>>1, we have from the energy equation such as shown in (6.19) CE*3/2*-* (6.24) Replacement of (6.24) to (6.21) and (6.23) gives the following relations: The coefficient * in (6.26) is about two times large compared with that given by Mellor, while the coefficient * in (6.26) is almost the same as Mellor's value.* If *=0, we have from (6.19) and (6.23) the following two relations: Thus we have Replacement of C1=1.5 and C2=0.65 reduces to E**2.90, *M*1.00 (6.30) The value of *H at *=0 is estimated by making use of the following turbulent Prandtl number Pr, which is derived from (6.21) and (6.23). In our case we have Pr=0.75, *H=0.75 (6.31) If we use the values of E*, *M and *H at *=0 mentioned above, we have at *=0 As mentioned above, the values of *M, *H, u*2, w*2, ** at *=0 (or z*0) are determined based on parameters of C1, C2, C1T, CT' * If C1=1.5, C2=0.65, C1T=3.2, C2T=0.5, CT' =1.6, we have *0.22 and *0.39. **In the neutral case the strong shear of mean flow exists we have u*2>**2>w*2. In this sense the application of our discussion is limited in the case the shear of mean flow is weak.

10 October 1978 Kanzaburo Gambo which are determined from the experiment of the homogeneous shear flow or the buoyant flow in the wind tunnel. On the other hand Wyngaad et al. (1974, 1975) proposed the similar equations for uiuj, uj* and *2 with ours, though the expression of *ij2 is the type of (3.15). However they used the observed boundary layer values of *M, *H, u*2, w*2, * at *=0 in order to determine the values of C1, C2, C1T, CT' In this sense, there is no guarantee that the govering equations for uiuj, Ui* and *2 proposed by Wyngaad et al. (1974) could be safely applied to the constant flux layer. According to numerical experiment of the planetary boundary layer, however, the result seems to be reasonable compared qualitatively with Deardorff's (1972) experiment. The comparison of their result with ours will be briefly explained in section Eddy coefficients far momentum and heat, KM and KH. The prognostic equations in the planetary boundary layer are given by (5.1)*(5.10) as mentioned in section 5. However, the prognostic equations for uiuj, ui* and *2 may be simplified if we assume that locally steady state for the quantities of uiuj, ui* and *2 is satisfied and their diffusive processes are neglected. Mellor and Yamada (1974) classified such a system as the level II in their terminology. In this case the equations for w2, uw, u*, w* and *2 may be written, refering (5.8), (5.9) and (5.10), as follows: Here we assume that Ui= {U, 0, 0} and U=U(z). The energy equation for E=(u2 +v2 +w2)/ 2 is given by After some algebraic manipulation the solutions of the set of equations (7.1) to (7.6) may be expressed as follows: KM and KH are the eddy coefficients for momentum and heat and Rf is the flux Richardson number, i.e., In our case we have Rf1*0.29, Rf2*0.33, Rf 3*0.45 Thus the critical flux Richardson number Rfc is

11 476 Journal of the Meteorological Society of Japan Vol. 56, No. 5 given by Rfc=0.29. Concerning the value of Rfc, Yamamoto pointed out that it depends on the functional form of *M and *H with respect to and showed that we may be able to expect * the non-existence of Rfc in a special case. In our case the value of Rfc=0.29 is close to those suggested by Businger et al. and Kamimal and Izumi (1965). Since the functional forms of KM and KH in (7.18) and (7.19) are complicated ones, Takano (1977) rewrite (7.18) and (7.19) in the following way Here we use the relation such as And a, b2(ri) and b2(ri) are computed refering the results obtained by Mellor and Yamada. These formulas for KM and KH are originally derived by Yamamoto and he finds the empirical constant values of b1=25, b2=40 and a=1.25, under the assumption that b1(ri)=const., b2(ri)= const.. In order to compare our result with those obtained by Takano and Yamamoto, we discuss the case -*>>1. When -*>>1, we have from (7.20) and (7.21) because b1(ri) and b2(ri)>>1, On the other hand Mellor and Yamada obtain the following approximate expression (refer Fig. 1 in their paper): If we compare (7.24) and (7.26) with (7.28) and (7.30) the values of KM and KH estimated by Mellor and Yamada are about two times large compared with ours for given value of Rf. If we use the empirical constants of a=1.25, b1=25 and b2=40 in (7.22) and (7.23) suggested by Yamamoto, we have These formulas correspond to (7.29) and (7.31). Our results of KM and KH are larger than the values estimated by Yamamoto about 10% and 25% respectively when -Ri>>1. 8. Scale-length l The eddy coefficients for momentum and heat, KM and KH are expressed with the parameter l as shown in (7.8) and (7.9). In the constant flux layer we have l*kz outside the roughness sublayer. If we consider to apply the eddy coefficients, KM and KH to the planetary boundary layer, we must stipulate the function l(z). In the present paper, we assume Blackadar's (1962) interpolation formula l=kz/1+kz/l0 (8.1) Here we use Rf*1.446 Ri. In our case we have from (7.16) and (7.17) Therefore the approximate expressions for KN and KH in (7.18) and (7.19) are given by which interpolate between two limits l*kz as z* 0 and l*l0 as z*. Various expressions for l0 are proposed. For example, l0=ug/ U9 is the geostrophic velocity or l0=u*/f in the neutral boundary layer. However it is shown that these representations are not satisfactory in the convective turbulence field (Deardorff, 1974a), In the present paper we propose the following form for l0:

12 October 1978 Kanzaburo Gambo 477 l0=h/n (8.2) Here we note that h is the height of convective layer and N is a constant to be determined. Roughly speaking, N corresponds to the value of 1/* which is defined by Mellor and Yamada (cf. (7.2) in their paper.). Now we consider to determine the appropriate value of N. Introducing the expression of (8.1) into (5.8), (5.9) and (5.10), refering the discussion of the constant flux layer in section 6, we have the following equations corresponding to the equations (6.4)*(6.10): The energy equation is written from (8.3)*(8.5) as follows: In a strongly unstable case (-*>>1), the equations (8.3)*(8.11) are simplified just as we computed the values of *M and *H in section 6. After same algebraic manipulation we have * is given by (6.27). On the other hand we have from (8.11) Then we have from (8.9) *= *CT'/C2/3.* Eliminating *M from (8.11) and (8.5), we have In order to compare with the observational data in the planetary boundary layer, we compute the following values: * In our case *l.02 and *2.83

13 478 Journal of the Meteorological Society of Japan Vol. 56, No. 5 w* is the convective velocity scale defined by Then we have k is the Karman's constant. Assuming that we compute W2 and *2 for N=5, 10 and 15 and the results are shown in Figs. 7 and 8. For the sake of comparison, the observational data in the figures are quoted from Deardorff's (1972) paper. In the figures the results of numerical experiment in a case of -h/l=45 computed by Deardorff (1972) are also shown by the dotted line, although his example corresponds to the case w*/w*0=(1-z/h)2. Comparing the values of W2 with the observational data in Fig. 7, the value of N seems to be around N=5*10 i.e., h/5*10. the differences In the case of *2 in Fig. 8, however, l0* of *2 due to the value of N is rela- Fig. 7 Mean profiles of vertical velocity variance, W2=w*2u*02/w*2 for N. Ordinate is z/h (h: height of convective layer). Observed data are quoted from Deardorff's (1972) paper. Dotted curve is obtained by Deardorff (1972) for h/l= Fig. 8 Mean profiles of potential temperature variance, *2=**2(W*2/u*02) for N. Ordinate is z/h (h: height of convective layer). Observed data are quoted from Deardorff's (1972) paper. Dotted curve is obtained by Deardorff (1972) for -h/l=4.5 tively small and it is difficult to determine the reasonable value of N from the observational data. Here we note that Wyngaad et al. (1974) obtained the similar results with those in Figs. 7 and 8. In their case the prognostic equation for is used instead of defining the scale-length and * they determined the constants for the parametarization of closure model from the observational data at *=0 as mentioned in section 6. Their results w*/w*0*(1-z/h) and -z/l>>1* are shown in Figs. 9 and 10 by dotted line, while our results in Figs. 7 and 8 are shown by contineous lines. As may be seen in Figs. 9 and 10, the case of N=5 seems to correspond to their result in case of W2. In case of *2, however, N= 15 seems to correspond to their result. If we assume that N=10 and h*1,000m, we have l=0.4z/( z), i.e., l*67m at z= 500m. For the sake of comparison, we refer the observational result of the Eulerian scale length e obtained by Gamo et al. (1975). They showed l that the vertical distribution of le is quite similar to (8.1) and le is about 100m at z=500m* 1,000m. Though there is some problem about the direct comparison of l with 1e, our assumption that N*10 may be reasonable. Since l is proportional factor in estimating the eddy coefficients of momentum and heat, the numerical simulation of the planetary boundary layer should be per- * They computed the cases of -h/l=10, 20 and 50. The results show that there is no difference in each case.

14 October 1978 Kanzaburo Gambo 479 *w*31.7*1/3*** (8.22) Gamo et al. (1976) showed that *w=c0*1/3 and the mean proportional factor C0 which was determined from the observations at the heights of 200m, 300m, 460m, 700m and 1,000m is about 28, i.e., *w*100cm/sec, when *=50cm2/ sec3. Strictly speaking, C0 is regarded as C0= 1/2[l(z)]1/3 such as shown in (8.21). However, * Gamo et al. (1976) estimated the mean value of C0 in the planetary boundary layer. Therefore we omit the further discussion about the quantitative difference between the coefficient in (8.22) and the value of C0. Fig. 9 Comparison of W2 for various values of N (solid line) in case -*>1 with that obtained by Wyngaad et al. (1974) in case -h/l>>1. Ordinate is z/h. Acknowledgement The author wishes to express his thanks to Prof. T. Matsuno and Dr. K. Takano for their kind discussion. Thanks are due to Mr. Y. Fujiki and Mrs. K. Kudo for drawing the figures and typing the draft. The author is grateful to the reviewers for their valuable comments on the original version of this paper. The research reported here was supported by Funds for Scientific Research from the Ministry of Education. Fig. 10 Comparison N of *2 for various values of N (solid line) in case -*>>1 with that obtained by Wyngaad et al. (1974) in case -h/l>>1. Ordinate is z/h. formed in the future in order to check the vertica distribution of l comparing the observationa data. As a short comment to (8.15), here we refer the observations of w2 and * by Gamo et al (1976). The equation (8.15) is rewritten as fol lows: Here we use the relation of *(g/*0)w* uncler the assumption of -*>>1. Putting *2.82* amd l=67m**, we have in unit of cm/sec * cf. footnote of (8.15) N=10 and h=1,000m ** are used in estimating the value of l. l*67m is the value at z=504m. References Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in neutral atmosphere. J. Geophy. Res., 67, Businger, J. A., J. C. Wyngaad, Y. Izumi and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, Champane, F. H., V. G. Harris and S. Corrsin, 1970: Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech., 41, Crow, S. C., 1968: Viscoelastic properties of finegained incompressible tu*bulence. J. Fluid Mech. 33, Deardorff, J. W., 1972: Numerical integration of neutral and unstable planetary boundary layers. J. Atmos. Sci., 29, a: Three-dimensional *, numerical study of the height and mean structure of a heated planetary boundary layer. Boundary- Layer Meteor., 7, b: *, Three-dimensional numerical study of turbulence in an entraining mixed layer. Boundary-Layer Meteor. 7, Gamo, M. T. and O. Yokoyama, 1975: Observed characteristics of the standard deviation of the vertical wind velocity in upper part of the atmospheric boundary layer. J. Meteor. Soc. Japan, 53, *** *w123cm/sec when =50cm2/sec3.

15 *,O 480 Journal of the Meteorological Society of Japan Vol. 56, No. 5 from a simplified three-dimensional numerical model of atmospheric turbulence. J. Atmos. Sci., 33, Sommeria, G., 176: Three-dimensional simulation of turbulent processes in an undisturbed tradewind boundary layer. J. Atmos. Sci., 33, Takano, K., 1977: Three-dimensional numerical modeling of the land and sea breezes and the urban heat island in the Kanto Plain. Sc. D. Thesis, Tokyo University (to be published). Webster, C. A. G., 1964: An experimental study of turbulence in a density stratified shear flow. J. Fluid Mech., 19, Wyngaad, J. C., O. R. Cote and Y. Izumi, 1971: Local free convection, similarity, and the budgets of shear stress and heat flux. J. Atmos., Sci., 28, *,* and K. S. Rao, 1974: Modeling the atmospheric boundary layer. Advances in Geophysics, 18A, : Modeling *, the planetary boundary layer extension to the stable case. Boundary- Layer Meteor. 9, Yamada, T., 1975: The critical Richardson number and the ratio of the eddy transport coefficient cbtained from a turbulent closure model. J. Atmos. Sci., 32, and G. *, L. Mellor, 1975: A numerical model prediction of the Wyangara atmospheric boundary layer data. J. Atmos. Sci., 32, Yamamoto, G., 1976: Generalization of the KEYPS formula in diabatic condition and related discussion on the critical Richardson number, J. Meteor. Soc. Japan, 53, A. Shimanuki, *, 1966: Turbulent transfer in diabatic conditions. J. Meteor. Soc. Japan,