Growth of the Mixing Depth and the Diurnal Variation of Vertical Profiles of Temperature and Turbulence Characteristics in the Mixing Layer

* , eddy diffusivity Km, show the same tendency in each group. However, the temperature gradient shows the adiabatic lapse rate in both free and forced convective layer. This means that the unstable layer (d*! dz<0) is unobservable in most cases except the surface layer. This coincides with the fact that if there is a small amount of the surface heat flux when the wind is not so strong, free convection becomes easily predominant. Considering data obtained by airplane, Yokoyama et al. (1977a) proposed a theoretical

April 1979 M. Gamo and O. Yokoyama 159

Growth of the Mixing Depth and the Diurnal Variation of Vertical Profiles of Temperature and Turbulence Characteristics in the Mixing Layer

By Minoru Gamo and Osayuki Yokoyama

National Research Institute for Pollution and Resources, MITI, Tokyo 115 (Manuscript received 16 February 1978, in revised form 18 January 1979)

Abstract

A simple but practical model for the growth of a connective mixing layer is derived by integrating the entrainment rate equation proposed by Deardorff et al. (1969). The development of the mixing layer height is determined by three parameters which are the potential temperature gradient in the stable layer capping the mixing layer, integrated surface heat flux and initial value of the mixing depth. Also a model for the time-height variation of the turbulence structure in the mixing layer is proposed based on the above growth theory and turbulence structure model of the mixing layer presented by Yokoyama et al. (1977a). In this paper, the surface heat flux is assumed to be proportional to the insolation disregarding absorption by the atmosphere. The model is applied to data obtained by airplane over flat terrain in the vicinity of Tokyo, in March 1972, summarized by Gamo et al. (1976a). Estimation of the diurnal variation of the mixing layer structure including potential temperature, energy dissipation rate, rms of vertical wind fluctuations and turbulent eddy diffusivity seems to agree favorably with observations.

estimating MMD. 1. Introduction For deriving the model of the above subject, It is said that the mean temperature in the two models are required to be established: one troposphere decreases with an increasing height the mixing layer structure model, the other the by 0.5-0.6*/100m. That is, typically observed model for the growth of the mixing depth. atmosphere shows slightly stable stratification. First, we will see the turbulence structure of Generally, on sunny days, this stable layer is the mixing layer. Gamo et al. (1976a) observed replaced, from below, by a free convective mix- turbulence by airborne measurements and classi- ing layer caused by surface heating by the sun. fied the atmospheric boundary layer below the The mixing depth limits the extent to which inversion base into two groups: free and forced pollutants can spread vertically, and turbulence convective atmospheric boundary layer. It was quantities in the mixing layer determine the be- found that vertical profiles of turbulence quanti- havior of the dispersion. Also the mixing layer ties, such as the energy dissipation rate *, rms determine the behavior of the dispersion. Also of the vertical component of wind fluctuations the mixing layer structure is important for gen- eral circulation models, for the numerical weather prediction, etc. Thus, problems of predicting the variation of the mixing depth and turbulence characteristics in the mixing layer with time is of great practical concern. So far, for air pollution problems, the maximum mixing depth MMD has been predicted. It would be of great use if the development of the convective mixing layer and turbulence features in it were also able to predicted by a simple method such as that of 160 Journal of the Meteorological Society of Japan Vol. 57, No. 2 model for the free convective atmospheric applied to the 1972 Kawaguchi airborne measure- boundary layer. Assuming that buoyancy term ments (Gamo et al. (loc.cit)). and energy dissipation term are balanced at each level in all layers below the mixing depth, they 2. Development of the mixing depth hm during derived the mixing layer structure model which daylight hours gives profiles of heat flux Q, *, *w, rms of tem- If free convection is prevalent and air is mixed perature fluctuations *, and Km. This model is rapidly below the inversion base, the potential obtained by extending the Monin-Obukhov (1954) temperature through the whole mixing layer *m theory established in the constant flux layer. is expected to become constant, i.e., the tempera- According to this model, vertical profiles of d• This is in accord with observations * except turbulence quantities are described by two param- ture gradient becomes the adiabatic lapse rate eters: surface heat flux Q0 and the mixing depth for thin layers near the earth's surface and just hm. Furthermore, as hm is represented by the below the inversion base. In further discussion, integration of Q0, turbulence features can be for clarity and simplicity, we will assume that derived by only Q0 in implicit form. Next, we will look at the model of the growth of the mixing layer briefly. Ball (1960) clarified the concept of the mixing layer structure. Lilly is applicable for the whole mixing layer at any (1968) obtained the equation of the mixing layer time. If *m is increasing with time, i.e., the sur- development based on Ball's assumption. face heat flux Q0 is positive, this assumption Deardorff et al. (1969) established the successful means that the heat supplied from the surface formula for the mixing layer growth rate. is transported upward uniformly and the heating Tennekes (1973), Stull (1973, 1976), Carson rate of the air is independent of the height below (1973), Zilitinkevich (1975), Mahrt and Lenschow the mixing depth hm. So the heat flux Q decreases (1976), Zeman and Tennekes (1977) and others linearly with height z at any time as follows: developed the ideas of the mixing layer growth rate proposed by Deardorff et al. and Lilly, considering temperature jump, large scale subsidence, According to data obtained by Lenschow (1970), radiation, turbulent energy budget including Kukharets and Tsvang (1976), Yamamoto et al. shear-driven turbulence and turbulent energy diffusion, overshooting of plumes, the difference (1977) and others, Eq. (2) is verified as the first approximation. of initial conditions, etc. The variation of *m with time is described by It is expected that the diurnal variation of the the thermal equation for the mixing layer neglect- mixing layer structure can be obtained by com- ing the horizontal advection and radiation, as bining the equation for the mixing layer growth follows (e.g., Lilly (1968)): proposed by Deardorff et al. (loc.cit) and the mixing layer structure model proposed by Yoko- yama et al. (loc.cit). This is the main subject of the present paper. where Q(z) is the sensible heat flux at a height For the sake of clarity and simplicity, we dis- of z, p the mean air density in the mixing layer regard the horizontal advection, large scale sub- (~1.2*103g/m3), Cp the specific heat of air at sidence, the effects of latent heat and radiation, constant pressure (~0.24cal/gK). shear-generated turbulence, super adiabatic layer We can postulate that Q=0 at hm, because near the surface and temperature jump in the we assume that there is no temperature jump at vicinity of the mixing layer height. From the hm. Temperature jump seems to be slight or observational point of view, it seems that ac- unobservable above land on clear days when the curacy of empirical data obtained by a moving growth rate of hm is great. Considering that platform such as an airplane is as yet not suf- d*m/dt is independent of the height, integration ficient for comparison with more detailed estima- of Eq. (3) becomes tion considering the complicated phenomena mentioned above. For the purpose of obtaining the simplified first order approximation, we assume that Q0 is a linear function of insolation Next, assuming that the potential temperature at the top of the atmosphere I0. Our model is gradient of the stable layer capping the mixing April 1979 M. Gamo and O. Yokoyama 161

layer d*s/dz(=*s) is constant with height and layer in early spring. The summer case is shown time, the following expression can be obtained in Fig. 2. In both cases, the temperature gradient as shown by Deardorff et al. (1969), below the inversion base hi were most often *d or near *d, which means that free convection was predominant below hi. Airplane measurements were carried out only in fair weather, so Integrating Eq. (5), we can obtain the follow- results seem to represent the typical characteristics ing expression in clear skies in each season. There is a marked difference for *s; *s=0.0036*/m in early spring, 0.0049*/m in summer. Here *s is where suffix 0 shows the initial value. If *s is roughly calculated by the temperature difference known by the aerological data, Eq. (6) means between hi and the highest point in each flight. that we can predict the development of hm by measuring the temperature only at one level in the mixing layer using a tall tower or captive balloon, etc. From Eqs. (4) and (5), the equation of the development of the mixing depth hm can be obtained as established by Deardorff et al. (1969):

Differential equations (4), (5) and (7) are the simple but basic relationships between hm, Q and m in the mixing layer. * Integrating Eq. (7) from t0, the time from when Q0 becomes positive, we can obtain the mixing depth hm at time t, as follows:

Fig. 1 The temperature gradient in the stable layer capping the mixing Eq. (8) is the same as Eq. (8) in the paper of layer in March: 1-*d, 2-0.0036 Tennekes (1973) when there is no potential tem- /m. * perature jump at hm. Strictly speaking, we can select t0 (hm=h0 at t0) at any time after Q0 becomes positive. Eq. (8) shows that hm(t) is determined by two constant parameters; *s and , and the integration of one variable parameterh0 Q0(t), i.e., the heating history of the mixing layer. Here we note the three parameters briefly. It is said that *s is 0.3-0.4*/100m (i.e., temperature gradient is 0.5-0.6*/100m) in clear, steady weather, as mentioned in the introduction. It seems that the value of *s differs slightly in different places and seasons according to many observations. But as yet we do not know the systematic grouping of *s. As an example, we show the seasonal difference of *s. Observations were carried out by continuous flights of an instru- mented airplane (Cessna 207) above the Kanto Plain during 1974-1976. Temperature was meas- ured by the thermistor thermometer whose re- Fig. 2 The temperature gradient in the sponse time is about three second. The climbing stable layer capping the mixing rate of the airplane was about 3m/s. Fig. 1 layer in summer: 1-*d, 2-0.0049 shows the temperature gradient above the mixing /m. * 162 Journal of the Meteorological Society of Japan Vol. 57, No. 2

Fig. 4 Relations between the inversion base hi obtained by solar and the integrated insolation *I0dtsolidline-hi~(*I0dt)1/2. : Dataare from Gamo et al. (1976b): 1-24 July 1974, 2-25 July, 3-26 July, 4-27 July, 5-28 July, 6-29 July, 7-30 July, 8-31 July.

Fig. 3 Seasonal difference of the growth of the inversion base height hi: early spring, 1-8 Finally, we will see the characteristics of ho, Mar. 1975, 2-3 Mar. 1976, 3-4 Mar. 1976, from which hm(t) starts to develop. In Fig. 4 it 4-5 Mar. 1976, 5-6 Mar. 1976, 6-8 Mar. is shown that h0 is different and the starting point 1976; summer, a-25 July 1974, b-26 July of hi(t) is also different in each case. The value 1974, c-13 Aug. 1975, d-14 Aug. 1975, of h0 seems to have two physical meanings. One e-15 Aug. 1975, f-17 Aug. 1976, g-19 Aug. is the case when h0 is the surface nocturnal in- 1976. version height which has been formulated during Fig. 3 shows the variation of hi with time the night. Shortly after sunrise, the surface in- during daylight hours. Although the sunrise in version whose potential temperature is *i, is spring is 1 hour or so later than in summer, the eroded by free convection. This case is of great relevancy to air pollution. Using t0 as the time growth rate of hi is nearly the same in both from when Q0 becomes positive, and t' as the seasons. In spring hi continues increasing in time of the breakdown of the nocturnal surface early afternoon, but hi saturates around noon in summer. This means that the potential tempera- inversion, Eq. (8) becomes ture gradient above the mixing layer *s has an important effect on the development of the mixing layer. The surface heat flux Q0 is determined by the The other case is when strong wind is pre- temperature difference between the surface tem- dominant before and at sunrise, so that the perature and the air temperature adjacent to the surface. Many factors, such as the surface char- forced convection with neutral stratification (*d) acteristics, time, season, latitude, cloud amounts, is formed up to h0, and the free convection becomes predominant from sunrise. In this case etc. determine the temperature difference be- the surface heat flux Q0 must warm up the thick tween the earth's surface and the air. However, as yet there is no systematic experimental data atmospheric layer up to h0 immediately after about the diurnal variations of Q0. Here we sunrise. assume that Q0 is proportional to the insolation If we consider the daytime variation of hm at the outer side of the atmosphere. Fig. 4 over cities where the mixing layer has been shows the relationships between the inversion base formed during the night by the surface urban- rural temperature differences, h0 is the initial hi obtained by sodar and integrated insolation obtained by a Robitzsch bimetal pyrheliometer. mixing layer height in Eq. (8). We regard hi as the upper side of the strong echo in the chart paper. The value of hi(t) increases 3. Height-time variations of the turbulence with the root of the integrated insolation (in de- quantities in the mixing layer tail, see Gamo et al. (1976b)). The same results According to Yokoyama et al. (1977a), energy during morning hours were obtained by Mori- dissipation rate *, rms of vertical component of guchi et al. (1977). This seems to show that the wind fluctuations *w, rms of temperature fluctua- term including Q0 in Eq. (8) is valid in real atmos- tions *s and the eddy coefficient Km in the mix- phere. ing layer are described as follows: April 1979 M. Gamo and O. Yokoyama 163

urement mentioned above, Cl and *1(*) become as follows:

We expect that the more accurate functional forms of * and the values of C will be obtained where C*, Cw, C* and Ckm are the universal constants, *, *, *, *km, the universal functions, by further more deliberate field experiments. But in this stage, the profiles of the set of the turbu- (=1- z/hm) the height parameter. Here * W* lence quantities are as follows: and T* are the velocity and temperature scales proposed by Deardorff (1970) and expressed as follows:

where g is the acceleration of gravity, T0 the mean temperature (K) in the mixing layer. Assuming that Q=0 at hm in Eq. (2), the vertical profile of heat flux is described by

Q=Q0*. According to Eqs. (20)-(25), turbulence quanti- Yokoyama et al. (1977a) obtained *'s from ties below hm are represented by Q0, hm, g, T0, airborne measurements of turbulence (Gamo et al. Cp, p and z. Since g, T0, Cp and p are assumed (1976a)), as follows: to be constant up to the height of 1,000m or so, the profiles of the above-mentioned quantities are determined by two external parameters; Q0 and hm, and *(instead of z). Each values of C's obtained from the same Strictly speaking, Eqs. (20)-(25) do not predict airborne measurements by Yokoyama et al. is as the real atmosphere in thin layers both near the follows: surface and near the top of the mixing layer. In the vicinity of hm there appear many phenomena, such as temperature jump, gravity wave, over- Tennekes (1970) estimated average values of shooting of thermal plumes, abrupt change of , and * in the mixing layer as follows: *w wind speed and direction, etc. On the other hand, in the surface boundary layer the temperature w=0.6W*, *0=4.2T*. * gradient shows unstable (d*/dz<0), friction It is interesting that values of Cw and C* ob- velocity is not negligible, thermal plumes appear tained by Yokoyama et al. are similar to inuniformly by the inhomogeneity of the surface Tennekes' results. characteristics. Therefore, observed data of Q0 Next, we will consider the scale of turbulence. and hm may be a little different from those used From Hanna's equation (1968), the eddy co- in the theory. efficient is described by The definitions of hm and Q0 by Yokoyama Km=0.09*wkm-1 et al. (1977a) are as follows: hm is the height where km is the wave number at which kSw(k) extrapolated linearly upward using two or more data in the middle of the mixing layer to the (Sw(k) being the vertical velocity spectral density at a wave number k) is a miximum. From Eqs. level where Q (or *) becomes zero. Q0 is the surface heat flux extrapolated linearly downward (11) and (13), a turbulent Eulerian scale le to the surface z=0 in the same way as hm. The (=1/2*km) is described by region where Yokoyama et al.'s model is applicable, seems to be in the mixing layer excluding 10 or 20% of the highest and lowest part of the mixing layer. If we use data obtained from the airborne meas- From this point of view, this model is a first 164 Journal of the Meteorological Society of Japan Vol. 57, No. 2 order model. However, Q0 is regarded as the realistic representative value above the lowest layer (0.1hm-0.2hm), because horizontal inhomogeneity above the surface layer seems to become slight. On the other hand, if in the vicinity of the inversion base there was none of These equations were already suggested by Yoko- the complicated phenomena mentioned above, yama et al. hm should correspond to the real inversion base Eqs. (20)-(24) describe the turbulence quanti- hi. Fig. 5 shows the comparison of hm defined ties at any time in the mixing layer. On the by Yokoyama et al. with hi estimated by tem- other hand, Eqs. (27)-(31) show values along the time axis. If we assume that hm is equal to the perature profiles by Gamo et al. (1976a). The mixing layer height hm defined by Yokoyama et inversion base height hi as the first order approxi- al. (1977a) is a little larger than hi in this obser- mation, the mixing layer structure is able to be vation. predicted by measuring temperature profiles only. Now, we will consider the relations between Strictly speaking, t* is necessary to be infinitesi- mally small when Q0 is changing with time. But Q0 and hm. These two parameters are not inde- the solutions of equations (27)-(31) are asymp- pendent, as seen in Eq. (7). Integrating Eq. (7) when the mixing layer depth develops from hm' totic, so there may be sufficient t* to analyze to hm in the time interval t*, when Q0 is regarded data. as a constant, it follows: Incidentally, there are relations between *, *w, and Km. From Eqs. (17)-(20), *we gain

Yokoyama et al. (1977a) obtained the above relation by the graphical method. Now, we replace equations (20)-(24) repre- This shows that if *0(t) is obtained from data sented by Q0 with equations described by the hm(t), other turbulence quantities can be calcu- time variation of hm (including *s and t*) in lated without using complicated equations (29)- terms of Eq. (26). It follows that (31). Here, we show relations between turbulence quantities at each level from equations (17)-(20), as remaining universal constants C's are unknown, it follows:

These equations show that the values of C are able to be determined by hm and turbulence quantities only at one level in the mixing layer. It is convenient, because only one value of C is obtained from one flight in the paper of Yoko- yama et al. (1977a). 4, Comparison of the model with observed results The model of the development of the mixing Fig. 5 Comparison of the mixing layer height layer and the model of the diurnal variation of hm defined by Yokoyama et al. (1977a) the turbulence structure in the layer described with the inversion base height hi sum- in previous sections are tested by observational marized by Gamo et al. (1976a). data. April 1979 M. Gamo and O. Yokoyama 165

We assume that the surface heat flux Q0 is sunrise (= 2*(tc-tr)/*), tc the meridian time (min), proportional to the solar radiation at the top of t* sunrise time. Of course, I0 may be simply the atmosphere, for the first order approximation, represented by the sinusoidal function as given as follows: by Carson (1973). We apply the model to observations KAW- MAR-1972 summarised by Gamo et al. (1976a). where we assume a is constant for simplicity. Values of *, 1, t*, tc, ts (sunset time) are as The reason for using that simplified assumption follows: is that we did not observe Q0(t), and we do not KAW-MAR-1972 (we use data on 9th March): know categorized values of Q0 for various meteor- t*(6:01), tc(11:52), ts(17:43), * (-4*43'50"), ological conditions. Under this assumption hm(t) (0.99283). l and *m(t) continue to increase till sunset, because We select three different atmospheric conditions Q0 is positive during daylight hours. In real as following. Case (A) is when a slightly stable atmosphere the time when Q0 becomes positive is layer whose potential temperature gradient *s after sunrise. Coefficient a becomes small in the starts from the surface at sunrise. This case late afternoon, because the difference between seems to appear frequently in summer, because the surface temperature and adjacent air tempera- the surface inversion layer does not develop ture, which determines the variation of Q0, be- easily due to warm surface temperature during comes small, due to increase of air temperature. night. The time change of the mixing depth hm(t) In addition, in the late afternoon, the energy is estimated by putting h0=0 in Eq. (8). flow out of the mixing layer and radiation effect Case (B) is when the strong surface inversion can not be neglected, because the "strength of layer has been formed during night, and above the mixing layer" becomes weak. Therefore, this the inversion there exists the same stable layer assumption can not be applied to the early as in case (A). This case occurs frequently from morning and late afternoon. However, this late fall to early spring on clear and calm days. assumption clarifies the seasonal and geographical The value of hm(t) is calculated by Eq. (9). differences in latitude. Also we will be able to Case (C) is when forced convection is pre- obtain a simplified and understandable solution. dominant until sunrise caused by strong wind, This is the solution of the extreme case. So we and the potential temperature gradient shows the expect the condition of the real atmosphere will adiabatic lapse rate *d . Then forced convection be obtained by some simple variation of this stops and free convection becomes prevalent just solution. after sunrise. In this case hm(t) is estimated by If we use the assumption Eq. (32), the time Eq. (8). variation of profiles of turbulence quantities can In analysis, the potential temperature gradient be obtained by equations (20)-(24) described by above the inversion layer *s, that of the surface Q0(t) or by equations (27)-(31) described by the inversion layer *i, and h0; the initial height of variation of hm(t). If the latter equations are hm are taken as follows: used, a sufficient time interval t* for calculation (A) *s=0.0033 (*/m) seems to be one hour or so in the afternoon, (B) *i=0.0423 (*/m) (we use data on 7th but in the morning it is necessary to take a March 1972) the height of the surface shorter time interval t*, because proper time inversion=200m interval t* is related to the growth rate of hm. s=0.0033(*/m) * We calculate insolation I0(t) by the following (C) *s=0.0033(*/m), h0=500m. equation Forced convection was predominant in the morning hours in the observation KAW-MAR- 1972. Strictly speaking, we can not compare observation with calculation, because calculation is made by assuming that free convection be- where I0 insolation at the top of the atmosphere comes predominant from the sunrise. But, fortu- (ly/min), S0 the solar constant (1.98 ly/min), * nately, turbulence characteristics (e.g., *, *) after latitude (Tokyo; 35*39'16"),* declination, 1 late morning are relatively insensitive to the initial the ratio of the distance between the sun and conditions, as mentioned later. So we assume earth with that average value, *=24*60(min), that observations selected by Yokoyama et al. hour angle (=2*(t-tc)/*),* *0 hour angle at (1977a) as a free convective layer can be corn- * , le and Km in three cases are as follows: 4.1 The mixing depth hm From Eqs. (8) and (32), when h0=0, we obtain

166 Journal of the Meteorological Society of Japan Vol. 57, No. 2

pared with calculations. Temperature profiles obtained by thermometers on the NHK broad- casting tower (Gamo et al., 1976b) show that there existed relatively strong surface inversion on the 7th, 8th and 10th March 1972. Runs of these days seem to correspond to the case of (B). There was a weak surface inversion on the 6th and 11th March. The height of the forced convective atmospheric boundary layer h defined by the extrapolation method (Yokoyama et al., 1976b) is 700-1,000m. But temperature profiles obtained by using an airplane (Gamo et al., 1976a) show that the inversion base height hi is two or three times less than h. These runs seem to correspond to the case of (C). We also assume that observed hi and potential temperature can be compared with the calculated ones even in the morning hours, because hi increases even when forced convection is predominant, and potential Fig. 6 Hourly variation of the mixing depth hm temperature indicates *d and increases below hi during daylight hours in cases of (A), (B) and (C). Numbers show runs when free (Gamo et al., 1976b). We can not understand convection is predominant which are selected why hi and the potential temperature increase by Yokoyama et al. (1977a). even in conditions of the forced convection. It seems that it might be because there is an input If a=0.2, the proportionality coefficient of Eq. of heat flux from the surface in the daylight hours even if forced convection is predominant. (35) becomes similar to that of Eq. (34). Busch et al. (1976) specified in his calculation that the It seems that free convection prevails above the maximum heat flux associated with sinusoidal forced convective layer, because the wind shear function of Q0 is about 18% of the solar constant. becomes smaller with increase of height (Gamo We use a=0.2 also in cases of (B) and (C) for et al., 1975). In this case heat flux seems to simplicity and easy comparison. The time de- decrease linearly with height in all the layers below hi, because the temperature gradient shows pendence of hm in the three cases are shown in Fig. 6. Development of hm are different in the d in both the forced and the free convective* three cases in the morning, but in the evening layers. Therefore, the time variation of the height the difference among them becomes small. It is of the upper free convective layer which cor- interesting that although h0 is 500m in case of responds to hi and the potential temperature (C), the height difference of hm between case (A) seems to be able to be described also by Eq. (26) (h0=0m) and (C) becomes only 100m after 10 and (4). o'clock. The same tendency is shown in case of Time dependence of hm, profiles of *, Q, *w, (B). That is, Eq. (8) rapidly loses its dependence on the initial conditions. Tennekes (1973) already suggested that this reason is because the entrainment is controlled by the dynamics of the turbulence and not so much by the initial conditions. Numbers written in Fig. 6 are run numbers when free convective mixing layer is formulated, which We select a coefficient a which leads nearly to the were selected by Yokoyama et al. (1977a). As observed data of * using Eqs. (34) and (21) or mentioned before, it seems that runs 3, 4 (6th (28) by the trial and error method. As a result, calculation is made when a=0.2, that is, 20% March 1972) and 22 (11th) belong to case (C), of the solar radiation is replaced in the surface and runs 8 (7th), 14 (8th) and 16, 17, 18 (10th) belong to case (B). heat flux. Gamo et al. (1976a) obtained empirical formula 4.2 Potential temperature in the mixing layer

The diurnal variation of *m in case of (A) is April 1979 M. Gamo and O. Yokoyama 167

Fig. 9 Observed hourly variation of *m when there exists nocturnal surface inversion (7 Mar. 1972) : dashed line, data by airplane; solid line, data by tower. Data are from Gamo et al. (1976a). Fig. 7 Hourly variation of the calculated potential temperature *m in the mixing layer and the mixing depth hm in case of (A).

Fig. 8 Hourly variation of the calculated Fig. 10 Hourly variation of the calcu- m and hm in case of (B). * lated *m and hm in case of (C).

shown in Fig. 7, where *0 at sunrise is taken afternoon. The time of breakdown of the as zero for simplicity. The variation of *m(t) is nocturnally established inversion is about three shown when there exists the surface inversion in hours after sunrise, which is similar to the O'Neill Fig. 8. Here, *m(t)=*ihm(t) while the surface experiments summarized by Carson (1973). In- inversion layer is being eroded, and after break- crease of the potential temperature from sunrise down of the surface inversion *m(t) is calculated to sunset is 5.5* in case of (A) and 13* in case of (B). On the other hand, in case of (C) by 200*i+*shm(t). Observed data of potential the potential temperature difference during the temperature on 7th March when a very strong day is only 4* (Fig. 10). surface inversion formulated during night are shown in Fig. 9. Agreement between observed 4.3 Energy dissipation rate * and heat flux Q and calculated *m is favorable except in the late The diurnal variation of * and Q are shown 168 Journal of the Meteorological Society of Japan Vol. 57, No. 2

Fig. 11 Hourly variation of the energy dissipation Fig. 13 Hourly variation of * and Q in rate * and heat flux Q in case of (A): solid case of (C). line, calculated; dashed line, observed values in the same day ; * : 200m, * : decreases rapidly with decreasing of Q0. In these 300m, * : 460 m, * : 700m. Observed data figures are also shown observational data of * are those for the cases in which free con- obtained by KAW-MAR-1972 experiment, all vections are predominant (Cf. Yokoyama et al 1977a) plotted data are those for the cases of the free convective conditions by Yokoyama et al. (1977a). Due to scant morning data, it is difficult to say which case agrees with our classification A, B and C. Although the scatter of data is consider- ably large, trend of the diurnal variation and values of * seems to correspond to estimation. 4.4 *, Q and hm diagram Figs. 14-16 represent diagrams of *, Q and hm in the three cases. Such forms are made by the sinusoidal variation of *0 or Q0, and con- tinuously increasing function of hm(t). In case of (A), the inclination of profiles of * or Q in the morning hours is almost the same in each case, that is, hm(t) is proportional to Q0(t). As Q0(t) increases approximately linearly with time from sunrise to about noon, hm(t) is proportional to hours from sunrise, such as proposed by Tennekes Fig. 12 Hourly variation of * and Q in (1973). On the other hand, inclination of *(z, t) case of (B). or Q(z, t) becomes gradually steep in the afternoon, by the decrease of Q0 and slow increase of in the same figures, because we assume that hm(t). buoyancy production term is balanced with the Fig. 15 indicates that the surface heat flux is energy dissipation term locally at any level below used at first for eroding the nocturnal surface hm. In Figs. 11-13 are shown time variations of inversion. Immediately after breakdown of the profiles of * and Q in three cases. Profiles of inversion, hm abruptly develops due to large and Q are different in the three cases in the* surface heat flux whose value is similar to that morning hours, such as in hm(t). But the shape at noon. This sharp change of hm(t) by break- and value becomes similar in the afternoon. That down is often observed in the chart paper in is, with increase of height, the peak values of solar observations. The diurnal variation of and Q appear at later time, and these values* inclination of * or Q varies markedly compared April 1979 M. Gamo and O. Yokoyama 169

Fig. 14 The diagram of *, Q and hm in case of (A). Fig. 16 The diagram of *, Q and hm in case of (C).

Fig. 17 Hourly variation of *w in case of (A): solid line, calculated; dashed line, observed. Symbols are the same as in Fig. 11. Fig. 15 The diagram of *, Q and hm in case of (B).

with case (A). Contrary to case (B), hm(t) rises slowly in case of (C) due to the warming up to the thick layer from the early morning when Q0 is yet small. As a result, hm(t) can not develop so much higher than in case of (A) or (B). 4.5 Rms of vertical wind fluctuations *w Semidiurnal variation of profiles of *w are shown in Figs. 17 and 18 in cases of (A) and (B). Fig. 18 Hourly variation of *w in case Case (C) is similar to case (A) except in early of (B). morning (see Gamo and Yokoyama (1977)). Profiles of *w are different in the three cases in velocity for the mixing layer W* has the peak the morning hours, but the shape and value be- value in the middle afternoon, because W* is comes similar in the afternoon. The scaling represented by sinusoidal function Q0 and con- 170 Journal of the Meteorological Society of Japan Vol . 57, No. 2

tenuously increasing hm. Therefore, *w(= CwW**w(*)) also has the peak value in the middle afternoon. With increase of height, the peak value of *w appears at later time. Stull (1976) estimated the time variation of W* whose value is within 20% of the observed values (O'Neill experiment). The scaling velocity W* obtained by Stull also has the peak value in the middle afternoon and value of W* is similar to our results. 4.6 Rms of temperature fluctuations * and Eulerian scale of turbulence le As * and le are proportional to *-1/4, their profiles are shown in the same figure. Fig. 19 represent *(t) and le(t) in case of (A). Values Of * and le become infinite at the height of hm by the function *-1/4. This curious phenomenon is caused by the inhomogeneity of temperature Fig. 19 Hourly variation of * and Eulerian scale at hm. Coincidence of calculation with observa- of turbulence le in case of (A): solid line, calculated; dashed line, observed. Symbols tion is poor. It suggests that the functional form are the same as in Fig. 11. of *(*) must be improved. 4.7 Eddy diffusivity Km Diurnal variations of Km which are independent of height in the three cases are indicated in Fig. 20. The time when Km has peak value occurs later than in *w due to the effect of hm2*1/3 in Eq. (31) or hm4/3 in Eq. (24). Observed data which is the average value below hm are also plotted in Fig. 20. Data of eddy diffusivity Km obtained by Sato (1977) from constant-volume balloon flight re- plotted in Fig. 20. Although values of Km are Fig. 20 Hourly variation of the eddy diffusivity Km different, observed Km also has peak value in the in cases of (A), (B) and (C). Numbers show middle afternoon, and it seems Km increases run when free convection is predominant slowly before the peak time and after that time in the KAW-MAR-1972 experiment. Data decreases rapidly till sunset. are from Yokoyama et al. (1977a). Eddy diffusivity Km obtained by Sato (1977) from 5. Conclusions constant-volume balloon flights: a-3-7 Mar. The model of the variation of the turbulence 1976, b-12-16 Oct. 1974, c-28-29 Nov. 1974, d-14-15 Dec. 1974. quantities with time during daylight hours is derived by combining the equation of the mixing layer development with the model of turbulence Third, the initial mixing layer height is 500m structure of the mixing layer. Calculation is at sunrise. Differences of hm(t) in the three cases made by a simple analytical method. The model are small, at most 200m. But differences of *m is applied to airborne measurements. It seems are remarkably large. that the mixing layer structure is reasonably ac- 2. Difference of turbulence quantities in the curately predicted by the present model. The three cases is large in the morning hours, but main results are as follows: becomes small in the afternoon, which means 1. Three different atmospheric conditions are that turbulence characteristics are relatively in- investigated. First, the stable layer capping the sensitive to initial conditions. mixing layer starts from the earth's surface at 3. The scaling velocity for the mixing layer sunrise. Second, the nocturnal surface inversion W* has the peak value in the middle afternoon, layer develops up to 200m above the surface. because W* is described by sinusoidal function April 1979 M. Gamo and O. Yokoyama 171

Q0 and increasing function hm. Therefore, pro- eddy transport in the planetary boundary layer files of *, and Km also has the peak value in the using characteristics of the vertical velocity middle afternoon. spectrum. J. Atmos. Sci., 25, 1026-1033. Kukharets, V. P. and L. R. Tsvang, 1976: On parametrization of turbulent heat flux in unstable Acknowledgements boundary layer of the atmosphere. Isv. Atmos. For suggestions and discussion thanks to our Oceanic Phys., 12, 13-21. colleagues Susumu Yamamoto, Hiroshi Yoshi- Lenschow, D. H., 1970: Airplane measurements of kado, Masayasu Hayashi, Koji Kitabayashi and planetary layer structure. J. Appl. Meteor., 9, Tateki Mizuno who are the members of the 874-884. Lilly, D. K., 1968: Models of cloud-topped mixed Atmospheric Diffusion Laboratory, National layers under a strong inversion. Quart. J. Roy. Research Institute for Pollution and Resources. Meteor. Soc., 94, 292-309. The work in this paper was supported under the Mahrt, L. and D. H. Lenschow, 1976: Growth dyna- special fund for pollution control of the Ministry mics of convectively mixed layer. J. Atmos. Sci., of International Trade and Industry. 33, 41-51. Monin, A. S. and A. M. Obukhov, 1954: Basic laws References of turbulent mixing in the atmosphere near the ground. Tr., Akad. Nauk SSSR Geofiz. Inst., Ball, F. K., 1960: Control of inversion height by No. 24 (151), 163-187. surface heating. Quart. J. Roy. Meteor. Soc., 86, Moriguchi, M., T. Kitade and F. Kimura, 1977: A 483-494. prediction method for "LID". The 18th Con- Busch, N. E., S. W. Chang and R. A. Anthes, 1976: ference of Japan Society of Air Pollution , p. 397. A multi-level model of the planetary boundary Sato, J., 1977: Estimation of eddy diffusivity in layer suitable for use with mesoscale dynamic urban area. The 18th Conference of Japan Society models. J. Appl. Met., 15, 909-919. of Air Pollution, p. 399. Carson, D. J., 1973: The development of a dry in- Stull, R. B., 1973: Inversion rise model based on version-capped convectively unstable layer. Quart. penetrative convection. J. Atmos. Sci., 30, 1092- J. Roy. Meteor. Soc., 99, 450-467. 1099. Deardorff, J. W., G. E. Willis and D. K. Lilly, 1969: -, 1976: Mixed-layer depth model based on Laboratory investigation of non-steady penetra- turbulent energetics. J. Atmos. Sci., 33, 1268- tive convection. J. Fluid Mech., 35, 7-31. 1278. 1970: Convective velocity and temperature -, Tennekes, H., 1970: Free convection in the turbulent scales for the unstable planetary boundary layer Ekman layer of the atmosphere. J. Atmos. Sci., and for Rayleigh convection. J. Atmos. Sci., 27, 27, 1027-1034. 1211-1213. -,1973: A model for the dynamics of the Gamo, M. and O. Yokoyama, 1975: Observed char- inversion above a convective boundary layer . J. acteristics of the standard deviation of the verti- Atmos. Sci., 30, 558-567. cal wind velocity in upper part of the atmos- Yamamoto, S., M. Gamo and O. Yokoyama , 1977: pheric boundary layer. J. Meteor. Soc. Japan, Airborne measurements of turbulent heat fluxes. 53, 412-423. J. Meteor. Soc. Japan, 55, 533-546. S. -, Yamamoto and Y. Mitsuta, 1976a: Yokoyama, O., M. Gamo and S. Yamamoto, 1977a: Structure of the atmospheric boundary layer On the turbulence quantities in the atmospheric derived from airborne measurements of the mixing layer. J. Meteor. Sac. Japan, 55, 182- energy dissipation rate *. J. Meteor. Soc. Japan, 192. 54, 241-258. -, 1977b: On the turbulence quantities in T. -, Mori and O. Yokoyama, 1976b: Ob- the neutral atmospheric boundary layer. J. Meteor . servation of the diurnal variation of the inversion Soc. Japan, 55, 312-318. base height. Pollution Control (Kogai), 11, 231- Zeman, O. and H. Tennekes, 1977: Parameterization 246. of the turbulent energy budget at the top of the -, and O. Yokoyama, 1977: On the diurnal daytime atmospheric boundary layer. J. Atmos. variation of the thickness of the free convective Sci., 34, 111-123. mixing layer and turbulence quantities in the Zilitinkevich, S. S., 1974: Resistance laws and pre- layer. Pollution Control, 12, 259-276. diction equations for the depth of the planetary Hanna, S. R., 1968: A method of estimating vertical boundary layer. J. Atmos. Sci., 32, 741-752. 172 Journal of the Meteorological Society of Japan Vol. 57, No. 2

混合層高度の発達と混合層内部における温度および乱流統計量の鉛直プロフィルの日変化について

蒲生稔・横山長之通商産業省公害資源研究所

Deardorff他(1969)により提案された式を積分することにより,単純で実際的な混合層高度の発達式を求めた。混合層高度の発達は混合層より上部の安定層の温位勾配,地表面における乱流熱輸送量の積分値および混合層高度の初期値の3パラメータにより決定される。また,上記混合層発達式と横山他(1977a)により得られている混合層の乱流構造式を基にして,混合層中における乱流統計量のプロフィルの時間変化を表わす式を求めた。本論文では,地表面熱輸送量は大気外日射量に比例すると仮定した。これらの式を蒲生他(1976a)により1972年3月に東京付近の平坦地形上で得られた飛行機観測データに適用してみた。混合層中における温位,粘性消散率,風速の鉛直成分の標準偏差,拡散係数の時間変化の計算値は観測値をかなり良く説明する。