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Cellular Cumulus Convection in a Moist Atmospheric Layer Heated Below* A ugust 1968 Tomio Asai 301 Cellular Cumulus Convection in a Moist Atmospheric Layer Heated Below* By Tomb Asai Geophysical Institute and Meteorological Research Institute, Kyoto University, Kyoto (Manuscript received30 January 1968, in revisedform 27 April 1968) Abstract The cumulus model proposed by Asai (1967) is applied to cellular cumulus convection in a moist atmospheric layer uniformly heated below and cooled above. The governing system of equa- tions derived has a steady state solution of cumulus convection independent of initial input distur- bances. A preferred mode of steady cumulus convection can be expressed in terms of an amount of heat supply, employing the selection principle that a preferred mode is one for which the lapse rate of temperature is the lowest. An examination of the results shows a coincidence with the feature of cumulus clouds in colder air over warmer water. However, tall vigorous cumulus convection is hardly expected to develop by uniform heating even with a large amount of heat supply. fragmental and little is known quantitatively of the 1. Introduction phenomenon described above. Further quantita- It has been well established that cellular convec- tive observations by aircrafts and meteorological tion, so-called Benard cell convection, takes satellites will be hopefully expected in the near place in a fluid layer heated below when the future to make clear characteristics of cumulus adverse temperature gradient exceeds a certain clouds and their distribution associated with en- critical value. This phenomenon has been con- vironmental conditions. firmed experimentally (Benard, 1900, etc) and An objective of the present article is to investi- theoretically (Rayleigh, 1916, etc.). gate theoretically what mode of cellular cumulus In the atmosphere, however, cumulus convection convection will prevail under the condition of a frequently occurs in a conditionally unstable layer given amount of heat supply from below. The because of the release of latent heat of water cellular cumulus model proposed by the present vapor. The situation mentioned above is usually author (1967) will be adopted in the following. observed over the warm ocean off the east coast In the previous study by Asai (1967), hereafter of the continents in winter, particularly when referred to as Paper A, there was assumed no fresh cold air outbursts (Asai, 1964, 1966; Ma- variation of static stability with respect to time tsumoto and Ninomiya,1965,1966, etc.). Another in spite of neither gain nor loss of heat in the example is a trade-wind cumulus which is often convection cell. Hence, the steady solution in observed in the tropics (Malkus, 195$ etc.). Paper A should be a trivial one, namely no Some observational studies made so far indicated convection, among two sets of solution (convection a close relation between cumulus convective acti- and no convection). Non-trivial solution (convec- vity and air-water temperature difference. This tion) discussed in Paper A implies that an enough suggests that heat (sensible and/or latent) supply amount of heat is gained and/or lost to maintain from the water to the air plays an important a steady state against convective upward heat role on cumulus convection. However, observa- transport. Thus the present investigation is re- tional data which have been obtained are still garded as another aspect of Paper A, that is, to * The work reported in this article was performed relate cumulus convection to an amount of heat at the Meteorological Research Institute, Tokyo, supplied from below. to which the author was affiliated. 302 Journal of the Meteorological Society of Japan Vol. 46, No. 4 the ratio of the updraft area to the whole area 2. Cumulus convection model which is simply called cloud cover later on. As Fig. 1 shows a simplified model for an axial is described in Paper A, a2 is a sort of entrain- symmetric convection cell: it consists of the as- ment constant which is assumed to be 0.1 in the cending motion in the inside column with the following. The first term on the right-hand side radius of a, the descending motion in the outside of the equation (1) is that of buoyant force pro- annular column with the radius of b, the inflow duced by the potential temperature difference be- in the lower layer and the outflow in the upper tween the inside and the outside columns <***>, layer. The cloud layer with the thickness of d, while the second term represents dissipation. which is filled by the convection cells, is assumed Detailed derivation of the equation (1) and its to be uniformly heated below and cooled above. physical significance are referred to Paper A. Now we assume that condensation of water vapor takes place only in the updraft and no evaporation occurs in the descending annular column surrounding the updraft. Thus we can obtain the thermodynamic equation for the po- tential temperature difference <***> referring to the derivation in Paper A. (2) where *d is the dry adiabatic lapse rate, *s is Fig. 1. A model of cellular cumulus convection. the moist adiabatic lapse rate and <***> denotes Applying the circulation theorem to the motion the difference between the average potential tem- in a meridional plane of the cell and making perature in the upper half of the cloud layer and some arrangements, we can obtain the equation that in the lower half. of motion for the updraft in the inside column Provided that an amount of heat flows into i.e., the cloud layer through the bottom and the same amount of heat flows out of it through the top (1) and furthermore vertical heat transfer in the cloud layer is accomplished only by cumulus convection concerned, then the thermodynamic equation for where the vertical difference of potential temperature may 'be obtained as follows (3) Here <wa> represents the updraft averaged over the inside column shown in Fig. 1. The ratio of a to b is denoted by *. Then *2 represents August 1968 Tomio Asai 303 where H denotes the upward flux of potential computations were performed. up to the period temperature at the top and the bottom of the of 90 minutes, which corresponds to 540 time cloud layer. Thus the first term on the right- steps, for a number of cases with different initial hand side of the equation (3) represents convec- conditions and parameters. tive heat transport at the mid-level of the cloud An example of results obtained is shown in layer, while the second term designates the heating Fig. 2, for which the following values of para- below and the cooling above. meters are used : The equations (1), (2) and (3) for <wa>, <d r *> and. <4z*1> will be solved for different values of d =1km, a/d =0.5 and *2=0.1. heat supply and of geometrical parameters *2, a Pressure and temperature at the bottom of the and d. cloud layer are assumed to be 950 mb and -5°C respectively. Temperature lapse rate is 8°C km-1. 3. Evolution of convection This situation is quite similar to the lower tropo- In order to investigate time-dependent charac- sphere observed over the Japan Sea coastal area teristics of convection governed by the system of of Japan in winter. Fig. 2 shows the evolution the equations (1) to (3), we made numerical inte- of the updraft *<Wa>*,the excess potential temper- gration of the equations with respect to time ature <4*r0> and the temperature lapse rate F*for using a revised Runge-Kutta method (Gill 1951). three cases with different initial input-disturbances A time step *t of 10 seconds was used. All and otherwise with the same condition. The solid line is for the case with the initial updraft of 1 m sec-1 and the dashed line for the case with the updraft of 0.1 m sec-1. No excess potential temperature is assumed for both cases. On the other hand the dash-dotted line is for the case with the excess potential temperature of 1°C and without any updraft at the initial moment. As is seen in Fig. 2, the same steady state can be attained for the three cases after about 40 minutes from the start, regardless of initial disturbances different from one another. This indicates that the system of the equations (1) to (3) has the solution which represents a steady state convection independent of an initial disturbance. 4. Steady state convection The steady state solution of the equations (1) to (3) such as shown in the previous section can be obtained as follows : (4) Note that the relationship <*zO>= (*d - F)d /2 Time (min) exists between * and <*z8>. The lapse rate * will Fig. 2. Evolution of updraft <*a>, excess poten- be used instead of <*z*> for illustrating the re- tial temperature <*r0> and lapse rate of te- sults obtained in the following, simply because mperature * for three initial disturbances di- the former is more familiar than the latter. fferent from one another. We now proceed to express a size of cumulus 304 Journal of the Meteorological Society of Japan Vol. 46, No. 4 Fig. 4. Lapse rate of temperature * against a fraction of the total area covered by the up- draft region *2. Others are the same as in Fig. 3. Fig. 3. Lapse rate of temperature * (°C km-1) against ratio of radius of updraft to its thick- air layer is heated below and becomes gradually ness a/d. Solid lines are for the warm air (20°C less stable. Similarly to the result obtained in at the bottom) and dashed lines for the cold Paper A, it is also found that the value of a/d air (-5° C at the bottom).
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