February

1982

T. Asai and I. Nakasuji

A Further Convection

425

Study of the Preferred Mode of Cumulus in a Conditionally Unstable Atmosphere By Tomio Asai

Ocean Research Institute, Universityof Tokyo, Tokyo 164 and Isao Nakasuji Orient Man Power System, K.K. (Manuscript received30 July 1981) Abstract Numerical experiments are made to determine a preferred mode of cumulus convection in a conditionally unstable atmosphere. The model developed in the previous study (Asai and Nakasuji, 1977) is extended to deal with water vapor explicitly. The preferred mode of cumulus convection is regarded as the steady convection cell attained eventually after a random potential temperature disturbance is imposed initially. It is shown that the preferred scale of the convection cell and the preferred cloud coverage depend on mean vertical velocity, static stability and relative humidity. It is confirmed that the preferred cumulus convection minimizes the potential energy and consequently the mean temperature lapse-rate in the convective layer.

1.

Introduction

The cumulus convection in a conditionally unstable atmosphere was studied by Asai and Nakasuji (1977), which hereafter will be referred to as paper AN, to see how the preferred mode depends on a mean vertical motion and a static stability in the atmospheric layer and the magnitude of each term of the energy equations depends on the cell size of cumulus convection. The results obtained are that the preferred cell size decreases and the preferred area ratio of the ascending region to the descending one increases as the mean vertical velocity increases. Without a mean upward motion the preferred cell size increases and the preferred area ratio decreases as the static instability decreases. While with a mean upward motion the preferred cell size decreases and the preferred area ratio increases as the static instability decreases smaller than a certain value. The preferred mode of a cumulus convection cell is the one for which the potential energy of the convective layer is at the lowest so that the mean temperature lapse-rate beccmes minimum. It was assumed in the previous paper that the

ascending motion was always saturated with water vapor while the descending motion was always unsaturated. This assumption may be allowed only when the convective layer is supplied with water vapor sufficiently. In the present paper water vapor is introduced into the model in an explicit form to examine the results obtained in the previous paper AN. 2.

Model

and formulation

of problem

2.1

Basic equations Consider a horizontally uniform atmospheric layer of the depth, h, which has a conditionally unstable stratification. Convective motions are restricted in the layer between two horizontall planes fixed at z = 0 and z = h, respectively. Horizontally averaged mean vertical motion does not necessarily vanish and the air can go through these two boundaries. The convective motion is confined to the vertical (x, z) plane. A pseudoadiabatic process is assumed, i.e., condensed water falls out of the system immediately. Ice phase of water is not considered. Then the conservation equations of momentum, heat energy, water vapor and mass for the shallow convection under the Boussinesq approximation can be ob-

426

tamed

Journal

as follows

(Ogura

of

the

and Phillips,

Meteorological

Society

of

Japan

Vol.

60,

No.

1

1962).

where T0 and p0 are the initial basic temperature and pressure, respectively. T00 and P00 are the temperature and pressure at the lower boundary, respectively. * is the constant temperature lapserate, R the gas constant of dry air and r the constant relative humidity, *0 and * 0 are obtained from T0 and p0 following their definitions. The following potential temperature disturbance is superimposed on the initial basic field.

where

(x, z) = aR(x, z) or *(x, z)=A(x)sin2(*z/h)

(2.13) * (2.14)

where *0, q0, *0, *, q and * are the initial basic values of potential temperature, specific humidity, pressure equivalent, *(p/ p0)(Cp-C*)/CP and their departures from the respective initial basic values. m, L, C and qs are the constant mean potential * temperature, the latent heat of condensation, the condensation rate and the saturation specific humidity. The other symbols are the same as in paper

AN

and

used

customarily.

Boundary conditions Both the upper and the lower boundaries are fixed and smooth for the convective motion while a uniform mean vertical motion can go through both the boundaries. The potential temperature and the specific humidity are assumed constant at the lower boundary, while constant fluxes of heat and water vapor are assumed at the upper boundary. The symmetrical conditions are adopted with respect to the lateral boundaries x = 0 and x = d. In summary,

where a is an amplitude of the initial potential temperature disturbance, b is its horizontal scale and R (x, z) is the two-dimensional random function which ranges from -0.5 to 0.5. (2.13) is adopted for determination of the preferred mode of convection while (2.14) for the other cases.

2.2

where *

is a uniform

d is the

horizontal

2.3

Initial The

initial

mean width

vertical of

the

velocity domain.

conditions basic

field

is set

up

as

follows,

and

3.

Computational

procedure

Introducing a stream function * and eliminating * from (2.1) and (2.2), we obtain a complete set of equations for *, * (= -* 2*), * and q. These differential equations are approximated by a set of finite-difference equations and solved numerically by the same method as paper AN. The variables are allocated to the respective grid points as shown in Fig. 1. The condensation rate C is computed by the same method as Asai (1965). We first estimate (t+ *t), q (t+ *t) and qs (t+ *t) by assuming * C=0 in (2.3) and (2.4). These tentatively estimated values of *, q and qs are denoted by **, q* and qs* respectively. When q0+q* defined as r - [iro8]o/h also becomes minimum at the preferred scale. The horizontal scale of convection cell which maximizes the total vertical heat transport (dashed line) is larger than the preferred scale. All terms of the kinetic energy equation (3.1) and the available potential energy

Fig. 5 Variations of magnitude of each term of the potential energy equation (4.3) with a horizontal scale of convection cell and number of occurrence of cell size, 1/h, for w=10 cm s-1, y=7 K km-1 and r=1. The scale for the potential energy is indicated on the right and for the other terms of (4.3) on the left. equation (3.2) have no maximum in the same range of the h rizontal scale of convection cell as in Fig. 5, though they are not shown here. Thus the selection hypothesis that the preferred convection maximizes any term of them is not supported. Fig. 6 is the same as Fig. 5 except for w=0,

Fig. 6 Same as Fig. 4 except and r=0.9.

for w=0,

y=9

K km-1

430

Journal

of the

Meteorological

' = 9 K km- i and r= 0 .9. In this case the convective layer is supplied with water vapor only by diffusive flux through the lower boundary. The frequency distribution of convection cell obtained for five different random disturbances with the same magnitude of the amplitude are contained in Fig. 6. A peak in the frequency distribution nearly coincides with the minimum potential energy and the minimum mean temperature lapse-rate, though the frequency distribution spreads to larger cell size. Thus the conclusion that the preferred mode of a cumulus convection cell is the one for which the potential energy of the convective layer is at the lowest is valid in the present model in which water vapor is introduced explicitly. 6.

Conclusions

Society

of

Japan

Vol.

60,

No.

1

increases and the preferred cloud coverage decreases in an atmospheric layer without a mean ascending current, while the preferred convection cell sligJitly changes its mode in the layer with a mean ascending current. The preferred convection realizes so as to minimize the potential energy and hence the mean temperature lapserate in the convective layer. The other selection hypotheses are not supported. These results are almost the same as those obtained by the previous paper. Acknowledgements The present work was financially supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. Most of the numerical calculations were made with the use of the FACOM M-1605 at the Ocean Research Institute, University of Tokyo. The authors are indebted to Miss M. Sobukawa for assisting computation and to Miss N. Takusa-

The model of the previous paper (Asai and Nakasuji, 1977) is revised to deal with water vapor explicitly. A preferred mode of cumulus convection is obtained by the same method as gawa for typing the manuscript. the previous paper. The results obtained are as References follows. The preferred scale of convection cell Asai, T., 1965: A numerical study of the air-mass decreases and the preferred cloud coverage intransformation over the Japan Sea in winter. J. Meteor. Soc. Japan, 43, 1-15. creases as the mean vertical velocity increases. The preferred scale increases and the preferred Asai, T. and I. Nakasuji, 1977: On the preferred mode of cumulus convection in a conditionally cloud coverage decreases as the initial basic unstable atmosphere. J. Meteor. Soc. Japan, 55, temperature lapse-rate ' decreases in the atmospheric layer without a mean ascending current, while in the layer with a mean ascending current this tendency of the variation is opposite so that the preferred scale decreases and the preferred cloud coverage increases. As the initial basic relative humidity decreases, the preferred scale

151-167. Ogura, Y., 1971: A numerical study of wavenumber selection in finite-amplitude Rayleigh convection. J. Atmos. Sci., 28, 709-717. Ogura, Y. and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sc., 19, 173-179.

条 件 付 不 安 定 大 気 中 に お け る積 雲 対 流 の 卓 越 モ ー ドに 関 す る 研 究 補 遺 浅

井 冨 雄 東京大学海洋研究所

中 筋 勲 株式会社オ リエント・マソパワーシステム 数値 実験 に 基 づ き,条 件 付 不 安 定 大 気 中 で 発 現 す る 積 雲 対流 の卓 越 モ ー ドの 力学 的 性 質 につ い て得 られ た結 果 が示 さ れ る。 前 論文(浅

井 ・中筋,1977)で

用 い られ た上 昇域 は常 に水 蒸 気 で飽 和,下 降 域 は未 飽 和 とい う仮 定

が と り除 か れ,水 蒸 気 を 陽 に 取 り扱 うモ デ ル に 修 正 され た。 前 論文 と同 様,最 初,温 位 に ラ ソダ ムな微 小 振 幅 の 擾 乱 を 与 え,最 終 的 に残 る 準 定 常 的 な 対 流 セ ル を 積 雲 対 流 卓 越 モ ー ドと した。 この よ うに して 得 られ た積 雲 対 流 の卓 越 モー ドは 対 流 セ ル の 水 平 規 模(あ

るい はaspect

ratio,即

ち 水平 規 模/鉛 直規 模)と 上 昇 流 域 の全 域 に対

February

1982

して 占め る割 合(あ

T. Asai and I. Nakasuji るい は 雲 量)に

431

よ って表 わ さ れ る。 これ ら卓 越 モ ー ドが 平 均場 の状 態 に い か に 依 存 す るか を

調 べ 大 略,前 論 文 の 結 果 を 確 認 す る こ とが で きた。 平 均 上 昇 流 が 増 す につ れ,卓 越 モ ー ドの 水 平 規 模 は 小 さ くな り,雲 量 は 大 き くな る。 一方 平 均 下 降 流 が 強 ま る と急 速 に水 平 規 模 は大 き く,雲 量 は 小 さ くな り,や が て 消滅 す る。 相 対 湿 度Vy対 す る卓 越 モ ー ドの応 答 は 平 均 上 昇 流 域 で は 鈍 感,平 均下 降 流 域 で は 敏 感 で あ る。 また,積 雲 対 流 の卓 越 モ ー ドは位 置 エ ネ ル ギ ー を最 低 に す る もの と し て実 現 して い る こ とも示 され る 。