A ugust 1968 Tomio Asai 301

Cellular Cumulus 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 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 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 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 of 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 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 , 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 **,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). Numerals labelled for cumulus convection at the lowest lapse rate at the lines indicate the amounts of heat sup- is almost independent of the parameters d and *2 plied below respectively in units of 100 cal and depends only upon a2. In the following, cm-2 day-1. therefore, the value of a/d is designated to be 0.5 for a preferred mode of cumulus convection. convection cell and its cloud cover, that is a, d Fig. 4 shows the variation of the lapse rate * and *2, in terms of an amount of heat supplied with the cloud cover *2 for a/d = 0.5 and d= 1km. below. Fig. 3 shows the variation of the lapse Solid, lines and dashed lines are for the warm air rate r against the ratio of the radius of an and the cold air respectively, in the same fashion updraft to its thickness a/d for *2= 0.1 and d = as in Fig. 3. Again there can be seen the lowest lkm. Solid lines are for the case of warm air lapse rates according to their respective amount characterized by 20°C at the bottom of cloud of heat supply. An increase in amount of heat layer, while dashed lines are for the case of supply shifts the position of the lowest lapse rate cold air characterized by -5°C at the bottom. toward a larger value of *2. Numeral labelled at each line denotes an amount Hence, we can determine a preferred mode of of heat supplied below in units of 100 cal cm-2 cumulus convection in terms of the amount of day-1. This diagram indicates that all the cases heat supply. Fig.5 (a) and (b) show the lapse have their respective lowest lapse rates at a/d = rate of temperature depending on the thickness 0.5 which are shown by small circles. It is of cumulus convection layer d and the cloud assumed here that a preferred mode of cumulus cover *2 for cases of the warm air (a) and of the convection is one which is maintained at the cold air (b), respectively. Contour lines indicate lowest lapse rate. The hypothesis posed above the lapse rate in units of °C km-1 for the amount seems to be reasonable for the situation that the of heat supply of 200 cal cm-2 day-1. Thus a August 1968 Tomio Asai 305

preferred mode of cumulus convection for which the lapse rate is the lowest value of 8.1°C km-1 can be determined to be d= 1.5km and *2=0.09 for the cold air. As for the warm air d=2.3km and *2=0.04 can be chosen as the preferred mode which corresponds to the lowest lapse rate of 5.5°C km-1. In this manner we can determine a preferred mode of cumulus convection for an amount of heat supply, as denoted by small circles for 100, 400, 600, cal cm-2 day-1 and so on. As is shown in Fig. 5 (a) and (b), both the thickness of cumulus convection layer and the cloud cover increase rather slightly with increasing amount of heat supply. For instance, the values of thickness remain at 2.2km for the cold air and at 3.0km for the warm air respecti- vely, even when an amount of heat as much as 1000 cal cm-2 day-1 is supplied. Also the values of *2 in this case are found to be 0.12 for the cold air and 0.06 for the warm air. Based on the combined analysis of heat budget and cloud ob- servation by aircraft, Matsumoto and Ninomiya (1966) and Matsumoto (1967) tentatively esti- mated d~2km and *2~0.10 for the amount of heat supply of about 500 cal cm-2 day-1. This was obtained over the Sea of Japan in the winter monsoon situation which was similar to the case of the cold air in the present paper. Most of estimates made so far on the sensible heat flux from sea to air are much less than 1000 cal cm-2 day-1 except for an extremely cold air outburst over warm sea. These are true particularly in the tropical regions. It may imply that some mechanism different from the uniform heating is required for larger-scale cumulus convection to develop.

5. Conclusions and remarks The cumulus model proposed by Asai (1967) was applied to cellular cumulus convection in a (b) moist atmospheric layer uniformly heated below Fig. 5. Lapse rate of temperature as a function and cooled above. The results obtained are sum- of the thickness of cumulus convection layer marized as follows. d and a fraction of the total area covered by (1) A steady cumulus convection independent the updraft region *2. Contour lines show the of initial input-disturbances is derived. lapse rate in units of °C km-1 for the amount (2) A preferred mode of cumulus convection of heat supply of 200 cal cm-2 day-1 for the warm air (a) and for the cold air (b). Small can be described in terms of an amount of heat circle indicates the mode of cumulus convection supply. Here we employed the selection principle at which the lapse rate is the lowest for the that a preferred mode of convection is one which respective amount of heat supply denoted by takes place and is maintained at the lowest lapse numeral labelled in units of 100 cal cm-2 rate of temperature. day-1. (3) The present theory is examined for the cold 306 Journal of the Meteorological Society of Japan Vol. 46, No. 4

air characterized by -5°C at the bottom of cu- advices and encouragement throughout this work. mulus cloud layer and for the warm air characte- rized by 20°C at the bottom. In the cold air References the thickness of cumulus convection d is 1~2 Asai, T., 1964: Photographic observation of clouds by km, the fraction of the total area covered by aircraft during snowfall period in Hokuriku Dis- the updraft region *2 is 812% and the lapse trict. J. meteor. Soc. Japan, 42, 186-196. rate * is 8.1~8.6°C km-1, while d=2~3km, *2 1966: Cloud bands over the . Japan -, Sea off =3~6% and *=5.5~6.1* km-1 in the warm the Hokuriku District during a cold air outburst. air, for the amount of heat supply ranging widely Papers in Meteor. and Geophys., 16, 179-194. from 100 to 1000 cal cm-2 day-1. These modes 1967: On the characteristics -, of cellular cumulus convection. J. meteor.Soc. Japan, 45, 251- of cumulus convection seem fairly well coincident 260. with the overall picture of the cumulus clouds Benard, H., 1900: Les tourbillons cellulaires daps une often observed over the oceans off the east coasts nappe liquide. Rev, gen. Sci. pur. appl., 11, 1261- of continents in winter and over the tropical 1271, 1309-1328. oceans. Still systematic observations are extre- Gill, S., 1951: A process for the step-by-step integra- mely lacking to examine quantitatively the feature tion of differential equations in an automatic dig- rather speculated here. We are now on the ital computing machine. Cambridge Phil. Soc. Proc., threshold of challenging this problem. 47, 96-108. (4) It is hardly expected that vigorous cumulus Rayleigh, L., 1916: On convective currents in a hor- convection with large values of d and *2 is pro- izontal layer of fluid when the higher temperature is on the under side. Phil. Mag., 32, 529-546. duced by uniform heating even with a large Malkus, J.S., 1958: On the structure of the trade amount of heat supply. This may suggest that wind moist layer. Pap. Phys. Ocean, and Met., Mass. the other mechanism is required for tall vigorous Inst. of Tech, and Woods Hole Ocean. Inst., 13, cumulus clouds to develop. 1-47. In the present model we assumed that water Matsumoto, S. and Ninomiya, K. 1965: An aeropho- vapor is supplied enough to maintain the cloud tographic observation of convective clouds in the layer saturated with water vapor against its de- vicinity of a cold dome center. J. meteor. Soc. crease due to condensation. It must be noted Japan, 43, 218-230. that an amount of water vapor supply may be a and- - , 1966: Some aspects. more important factor than that of sensible heat of the cloud formation and its relation to the heat supply in cumulus convection. and moisture supply from the Japan Sea surface under a weak winter monsoon situation. J. meteor. Acknowledgments Soc. Japan, 44, 60-75. Matsumoto, S., 1967: Some remarks on the convective The author would like to express his gratitude transfer under the north-westerly winter monsoon to Dr. K. Takahashi and Dr. S. Matsumoto, situation. Papers in Meteor. and Geophys., 18, 183- Meteorological Research Institute, Tokyo and to 191. Prof. R. Yamamoto, Kyoto University, for their

下か ら加熱 される湿潤な気層中における細胞状積雲対流

浅 井 冨 雄 京都大学理学部地球物理学教室,気 象学特別研究所

著者が前に(1967)提 案 した積雲モデルを一様に下面か ら加熱,上 面で冷却のある湿潤な大気層中で発現する細胞状 積雲対流の場合に拡張 して適用 した.こ の結果,運 動をひきおこすために最初に与え られた擾乱には依存 しない定常解 が得 られる.下 から加熱されて成層が次第に不安定化する気層中では最 も安定度のよい条件で発現維持 されるモー ドの August 1988 Tomio Asai 307

積 雲 対 流 が 卓 越 す る で あ ろ う とい う選 択 律 を用 い て積 雲 対 流 のモ ー ドを 下 か らの加 熱 率 に よ って 決 定 す る.得 られ た モ ドは 冬 期 の大 陸 東 岸 沖 や 又 熱 帯 海 洋 上 で観 測 され る積 雲 の 特 徴 と よい 一 致 を示 す.し か しな が ら大 き な激 しい積 乱 雲ー の 発 現 は こ こ で仮 定 され た一 様 な 加 熱 に よ って は 殆 ん ど期 待 し得 ない こ と も示 され る.