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

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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 predomi- nant. 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 tem- perature, 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 pollu- tion 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, con- sidering 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 as- sume 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 measure- ments 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 ob- tained 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.
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