# The effect on the evolution of the drop spectrum in clouds of the preconditioning of air by successive convective elements

S. Roesner, Andrea Flossmann, H. Pruppacher

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S. Roesner, Andrea Flossmann, H. Pruppacher. The effect on the evolution of the drop spectrum in clouds of the preconditioning of air by successive convective elements. Quarterly Journal of the Royal Meteorological Society, 1990, 116 (496), pp.1389 - 1403. 10.1002/qj.49711649607. hal-01905577

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# The effect on the evolution of the drop spectrum in clouds of the preconditioning of air by successive convective elements

By S. ROESNER, A. I. FLOSSMANN AND H. R. PRUPPACHER

Meteorological Institute, Joh. Gutenberg University, Mainz, F.R.G.

(Received 20 March 1989; revised 29 March 1990)

## SUMMARY

A multiple air-parcel model is used to study the effect on the drop size spectrum inside convective clouds of preconditioning of air by successive convective elements. Numerical evaluation of this model shows that each rising air parcel establishes a new environment for the air parcel following it, thus substantially affecting its thermodynamic behaviour. This preconditioning of air allows successive air parcels to penetrate an inversion and produce precipitation-sized drops under conditions which are not conducive to precipitation when only a single air parcel is considered.

## 1. INTRODUCTION

Earlier studies (Scorer and Ludlam 1953; Malkus and Scorer 1955; Scorer 1957; Woodward 1959, 1960; among others) have shown that the motion in convective clouds can be explained by assuming a series of rising thermals or air parcels which detach themselves successively from the heated ground and, because of their temperature excess, rise and at the same time continuously entrain drier and cooler air from the surroundings. This air-parcel approximation to the dynamics of atmospheric clouds has proved to be very useful in the past for describing the evolution of the drop-size distribution inside developing convective clouds. Whereas early models (Neiburger and Chien 1960; Warner 1969; Bartlett and Jonas 1972; Fitzgerald 1974) considered only the case of a single air parcel rising, and in addition assumed that the rising convective element behaved adiabatically, later models (Mason and Chien 1962; Warner 1973; Lee and Pruppacher 1977; Lee et al. 1980; Flossmann et al. 1985, 1987) considered more realistically the case in which air parcels entrain surrounding air as they rise. By including entrainment in the model it was possible to reproduce the typical double maximum often observed in number distributions of natural cloud drops. Parcel models with entrainment also lead to drop size distributions which are broader than the drop size distributions with a single maximum obtained from an adiabatic model. Recently, air-parcel models with entrainment have been extended successfully to include the effect of scavenging by nucleation and impaction. With this addition it became possible to follow the evolution in time of the size distribution of the aerosol particles remaining in the air, as well as the aerosol particle mass in the cloud water (Flossmann et al. 1985, 1987).

Several years ago, Mason and Jonas (1974) extended the single-parcel model to a double-parcel model in order to simulate a succession of bubble-type features in a convective cloud. In their model, two air parcels were assumed to rise in succession, whereby the second parcel rose through the air left in the wake of the first parcel after the first had reached its maximum height and then returned back to the condensation level in a downdraught. By comparing their model results with the observations of Warner (1969) the authors concluded that the drop size distribution of their model of the two parcels rising through each other was broader than that for the single parcel, and reproduced quite well the essential features of the drop size distribution observed by Warner. Unfortunately, however, Mason and Jonas (1974) considered only the effects
of condensation and did not take into account drop growth by collision and coalescence, which usually occurs quite early along with condensation growth during cloud development, and so they were not able to study the effects of successive air parcels on the development of precipitation-sized drops. They also assumed in their model, somewhat arbitrarily, that because of evaporative cooling, the first parcel would sink back to the condensation level before the second one rose. Although it is often observed that rising air parcels do indeed meet sinking parcels of penetrating downdraughts, this does not necessarily have to happen at condensation level: it may happen at every level in the cloud and it is just as likely that the rising air parcel meets the preceeding parcel without it descending, or even overtakes it while it is still ascending. Moreover, Mason and Jonas (1974) restricted themselves to a discussion of the effects of only two parcels.

The purpose of the present paper is to study the effect of the preconditioning of air, by a succession of rising convective elements, on the early evolution of the drop size spectrum and the formation of precipitation-sized drops in clouds by taking into account the mechanisms of condensation as well as those of collision and coalescence. This was achieved by reformulating our air-parcel model (Flossmann et al. 1985) to allow for the rise of three air parcels in succession through air of differing thermodynamic stabilities. The first of the parcels was allowed to ascend through clear air; the following second and third parcels then ascended through the cloudy air left in the wake.

We are keenly aware of the shortcomings of parcel models to describe the overall behaviour of convective clouds as compared, for instance, with a two- or three-dimensional cloud model. However, the parcel model, unlike these higher-dimensional models, does allow an explicit description to be made of the 'bubble' nature of a cloud arising from a series of convective elements and the effect produced on cloud microphysics; higher-dimensional models, on the other hand, consider a continuous updraught. In addition, most of these models have their microphysics parametrized, which thus precludes an explicit study of the evolution in time of the drop size spectrum in a cloud.

We are also aware of the criticisms which have been expressed in the literature regarding the treatment of entrainment in parcel models. Warner (1973, 1975), Manton and Warner (1982), Telford and Chai (1980, 1983), Telford et al. (1984), Telford and Wagner (1981), Baker et al. (1980), Baker and Latham (1979, 1982), Raymond and Blyth (1986), Hill and Choularton (1986) pointed out that, based on field observations, the entrainment process in convective clouds is not isotropic and homogeneous, and that air from the environment is not continuously and instantaneously mixed into the cloud, independently of location around the cloud as is assumed in parcel models. Also, entrainment is greatest at cloud top, in disagreement with widely used simple entrainment parametrizations. Therefore, these authors developed corrections to the entrainment formulations incorporating inhomogeneous and anisotropic mixing. Unfortunately, however, as pointed out by Jonas and Mason (1983), the improvements in these models were achieved only by the introduction of some new unrealistic features in the models. In addition, Hill and Choularton (1986) based their conclusions regarding the significance of an inhomogenous anisotropic mixing process on a study which included only the condensation mechanism and neglected the effects of collision and coalescence between drops which, in atmospheric clouds, begin already at an early stage.

For lack of further evidence, therefore, we employed in the present parcel model the concept of entrainment in its isotropic, homogeneous form, with the additional feature, however, that environmental air with fresh nuclei is entrained into the rising parcel only after that parcel had emerged from the air of the preceeding parcel. While inside the air 'left behind' by the previous parcel, this parcel was assumed to entrain only this residual air containing drops and interstitial aerosol particles. Thus, lateral
entrainment of environmental fresh nuclei is excluded during this stage. It will be shown later that this provision is responsible for the fact that the drop spectrum obtained by our model exhibits a double maximum only in the upper portion of the cloud, in agreement with field observations (e.g. Warner 1969).

## 2. THE PRESENT MODEL

The dynamics of the present multi-parcel model is essentially similar to that of a single-parcel model as formulated in Pruppacher and Klett (1978) and recently by Flossmann *et al.* (1985). For brevity the model equations will not be repeated here.

In order to describe a series of three air parcels, we made the following provisions in our model. Each of the three air parcels was assumed to start from just below cloud base having the same initial radius of 350 m, an updraught velocity of 1 m s⁻¹, and all with the same aerosol particle size distribution (see Eq. (3)). The temperature excess of the first parcel over the undisturbed environment was assumed to be 0.3 degC. The second and third parcels were assumed to have the same temperature excess but now with respect to the environment left behind by the previous parcel. While rising, the air parcels were assumed to entrain air of zero vertical velocity. However, heat, momentum, water vapour, aerosol particles and drops were assumed to be exchanged between the parcel and its environment.

In our model, each parcel was followed to its maximum height at which, through mixing, it came to rest. Since our model showed that each subsequent parcel achieved a vertical velocity higher than its predecessor, we formulated the model, for simplicity, in such a way that the parcels were prevented from overtaking each other during the ascent. Thus, the times of release of the second and third parcels were chosen so that these thermals would just pass through the previous one when it had stopped ascending.

The behaviour of the first parcel was assumed to be determined by the given temperature and moisture distribution of the undisturbed ambient air; and as the parcel was rising, its thermodynamic and microphysical parameters varied with height according to the model equations. These values for each height were taken to represent a new environment between the condensation level and the maximum height that the parcel reached and were then 'frozen' at each level and taken as the environment for the second parcel. This implied that at each level at which the second parcel arrived the values of the 'new' environment were read into the model, and in this way determined the motion and microphysics of the second parcel until this one had exceeded the maximum height reached by the first parcel. After this stage the undisturbed environment given by the initial conditions determined the behaviour of the second parcel for the continuation of the ascent. The same procedure was followed for the third parcel. This procedure assumed implicitly that in comparison with the cloudy parcel inside, the cloudy environment left behind by the parcel did not significantly change until the next parcel had passed through it. We considered that this assumption was justified given the fact that each successive parcel rose faster than its predecessor. Since the delay time for the release of the next parcel was set such that the parcel arrived at the time its predecessor had come to rest, the 'freezing-time' for the thermodynamic and microphysical parameters became rapidly shorter with increasing distance from the point of release.

The above dynamics was used to describe the variation in time of the number distribution function $f_d(m)$ and the mass density distribution function $g_w(m)$ for the drops formed in the parcels by condensation, collision and coalescence. These functions were determined, after Flossmann *et al.* (1985), from the relations:
$$ \frac{\partial f_d(m)}{\partial t} = -\mu(f_d(m) - f_d(m)')U + \left. \frac{\partial f_d(m)}{\partial t} \right|_{\text{act}} + \left. \frac{\partial f_d(m)}{\partial t} \right|_{\text{con/eva}} + \left. \frac{\partial f_d(m)}{\partial t} \right|_{\text{d,coal}} + \left. \frac{\partial f_d(m)}{\partial t} \right|_{\text{d,break}} \quad (1) $$

and

$$ g_w(m) = m f_d(m). \quad (2) $$

The mixing with the environment is characterized by the entrainment parameter $\mu = (1/m_a)(dm_a/dz)$, where $m_a$ is the mass of the air parcel, and $z$ is the vertical coordinate. This parameter $\mu$ is assumed to be 0.6/R, where R is the radius of the rising air parcel. $f_d(m)'$ denotes the drop spectrum of the environment and U is the updraught velocity. The contribution to Eq. (1) due to the activation of aerosol particles into becoming drops (act), due to the growth or evaporation of the formed drops by the diffusion of water vapour (con/eva), due to the collision and coalescence of drop (d, coal), and due to drop breakup (d, break) were computed from the detailed relations given by Flossmann et al. (1985).

The present model was evaluated numerically for a remote continental and a maritime atmospheric-aerosol particle spectrum. Following Jaenicke (1988) it was assumed that both aerosol types consisted of particles whose size distribution could be described by the superposition of three log-normal distributions:

$$ \frac{dN_{AP}}{d \ln r} = \sum_{i=1}^{3} \frac{n_i}{(2\pi)^{1/2} \log \sigma_i \ln 10} \exp\left(-\frac{\{\log(r/R_i)\}^2}{2(\log \sigma_i)^2}\right). \quad (3) $$

The appropriate parameters for the remote continental and the maritime distribution are given in Table 1. These parameters were tabulated by Jaenicke (1988) and are based on a large number of field measurements carried out by various investigators. The remote

<table><caption>TABLE 1. PARAMETERS FOR THE REMOTE CONTINENTAL AND MARITIME AEROSOL PARTICLE DISTRIBUTION AS GIVEN IN EQ. (3): $n_i$ = TOTAL NUMBER OF AEROSOL PARTICLES PER CM$^3$; $R_i$ = GEOMETRIC MEAN AEROSOL PARTICLE RADIUS (μm); $\sigma_i$ = STANDARD DEVIATION IN MODE i</caption><thead><tr><th>Mode</th><th>Size interval (μm)</th><th>Whitby mode</th><th>n<sub>i</sub></th><th>R<sub>i</sub></th><th>log<sub>10</sub> σ<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="3">Remote continental</td><td>1</td><td>0.001 &lt; r &lt; 0.1</td><td>nucleation</td><td>997</td><td>0.0010</td><td>0.328</td></tr><tr><td>2</td><td>0.001 &lt; r &lt; 0.1</td><td>nucleation</td><td>842</td><td>0.0218</td><td>0.505</td></tr><tr><td>3</td><td>r &gt; 1</td><td>coarse</td><td>0.00071</td><td>6.2400</td><td>0.277</td></tr><tr><td rowspan="3">Maritime</td><td>1</td><td>0.001 &lt; r &lt; 0.1</td><td>nucleation</td><td>133</td><td>0.0039</td><td>0.657</td></tr><tr><td>2</td><td>0.1 &lt; r &lt; 1</td><td>accumulation</td><td>66.6</td><td>0.1330</td><td>0.210</td></tr><tr><td>3</td><td>0.1 &lt; r &lt; 1</td><td>accumulation</td><td>3.06</td><td>0.2900</td><td>0.396</td></tr></tbody></table>

continental aerosol particles were assumed to consist of $(NH_4)_2SO_4$ with $\epsilon = m_s/m_N = 0.1$ in all modes, where $m_s$ is the mass of the water soluble ammonium sulphate and $m_N$ is the total mass of the aerosol particle. The maritime aerosol particle spectrum was assumed to consist of NaCl with $\epsilon = 1.0$ in all modes. The number density and volume density distribution of the aerosol particle spectra are given in Figs. 1(a, b) and 2(a, b). The specific gravity $\rho_{AP}$ of all aerosol particles was assumed to be 2 g cm$^{-3}$.

The temperature and humidity profiles in the atmosphere chosen for both the remote continental and maritime case are given in the skew $T - \log p$ diagram in Fig. 3.
<figure><img src="image_2.png" /><figcaption>Figure 1. (a) Number density distribution and (b) volume density distribution for aerosol particles of remote continental type (after Jaenicke 1988). The continuous curve results from the superposition of the three dashed curves; total number concentration $N_{AP} = 1366 \text{ cm}^{-3}$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. (a) Number density distribution and (b) volume density distribution for aerosol particles of maritime type (after Jaenicke 1988). The continuous curve results from the superposition of the three dashed curves; total number concentration $N_{AP} = 179 \text{ cm}^{-3}$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3. Vertical profile of temperature and humidity in the atmosphere assumed for the remote continental and the maritime case.</figcaption></figure>

### 3. RESULTS

#### (a) Effect of successive convective cloud elements on the drop size spectrum—remote continental case

The variation with height (i.e. with time) of the vertical velocity $U$, the liquid water content $w_L$, and the supersaturation $s_{vw}$ of the first, second and third parcels are displayed in Figs. 4(a, b, c) for the remote continental aerosol spectrum. We notice that each successive air parcel rises with a higher vertical velocity, and reaches a higher altitude than the previous parcel. This behaviour is summarized in Table 2 which shows that also the liquid water content reaches a higher value in each successive parcel. We notice, further, the discontinuities in the vertical variation of the supersaturation and liquid water content, for the second and third parcels. These discontinuities are the result of the different environmental conditions that a parcel experiences when it emerges from the environment of the air left by the previous parcel.

The supersaturation maximum which develops within the first 100 m above cloud base is the result in particular of the process of nucleation and condensation: as the air parcel reaches the condensation level it first loses the largest aerosol particles through activation. These, however, are only present in small numbers and thus are not capable of abstracting much water vapour from the environment to reduce the supersaturation significantly. Thus, the supersaturation keeps on rising until finally the large tail-end of the particle size distribution is activated and a large number of small drops are nucleated. At that moment the supersaturation decreases sharply as the numerous activated particles abstract substantial amounts of water vapour. The vertical velocity of the first parcel also decreases relatively soon after the maximum in supersaturation is reached, thus not causing a further rise of the supersaturation, through lack of cooling (see Fig. 4(a)). This is in contrast to the second and third parcels (see Figs. 4(b, c)) in which the super-
<figure><img src="image_5.png" /><figcaption>Figure 4. Variation with height and time of the vertical velocity *U*, the liquid water content *w*<sub>L</sub>, and the supersaturation *s*<sub>vw</sub>: (a) for the first, (b) the second and (c) the third parcel for the remote continental case.</figcaption></figure>

<table><caption>TABLE 2. VALUES FOR THE MAXIMUM VERTICAL VELOCITY *U*<sub>MAX</sub>, THE MAXIMUM LIQUID WATER CONTENT *w*<sub>L,max</sub>, AND THE MAXIMUM SUPERSATURATION *s*<sub>vw,max</sub> REACHED IN EACH SUCCESSIVE PARCEL FOR THE REMOTE CONTINENTAL CASE</caption><thead><tr><th>Parcel</th><th><i>U</i><sub>max</sub> (m s<sup>-1</sup>)</th><th><i>w</i><sub>L,max</sub> (g kg<sup>-1</sup>)</th><th><i>s</i><sub>vw,max</sub> (%)</th></tr></thead><tbody><tr><td>1</td><td>1.9</td><td>0.25</td><td>0.39</td></tr><tr><td>2</td><td>4.0</td><td>0.9</td><td>0.61</td></tr><tr><td>3</td><td>5.3</td><td>2.0</td><td>0.61</td></tr></tbody></table>

saturation, after its sharp decrease, rises again owing to continued cooling of the ascending parcel.

The variation in supersaturation is reflected in the drop-size number distribution function *f*<sub>d</sub>, given in Figs. 5(a, b, c). The first parcel only produces a narrow drop size distribution. Since the supersaturation does not rise after its low-level maximum, subsequent condensation cannot activate enough drops to produce a secondary maximum in *f*<sub>d</sub>, despite entrainment of new aerosol particles (Fig. 5(a)). Also the second and third parcels do not develop a secondary maximum in the drop size distribution as long as the parcels are inside the air left behind by the previous parcel. This is because, in the air left behind, there are only interstitial aerosol particles not activated during the ascent of the previous parcel, and their entrainment has little effect on condensation (Fig. 5b). As soon as the parcel emerges from the remnants of the previous parcel, new aerosol particles are entrained from the undisturbed environment which, owing to the rising supersaturation, are also activated to become drops. This causes a secondary maximum to develop in the drop size distribution. This behaviour is in agreement with field observations (e.g. Warner 1969) that drop size distributions exhibit double maxima which are most distinct in the upper part of a cloud. The same behaviour is also reflected in Fig. 5(c), which illustrates the drop size distribution in the third parcel.

The variation of the mass distribution function *g*<sub>w</sub>, which illustrates the location of the main water mass in the drop size spectrum, is displayed in Figs. 6(a, b and c). The
<figure><img src="image_6.png" /><figcaption>Figure 5. Variation with time of the drop number density distribution function $f_d$ for: (a) the first, (b) the second and (c) the third parcel for the remote continental case. ($N_d$: total number of drops). The distributions are given per logarithmic radius interval.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. Variation with time of the drop mass distribution function $g_w$ for: (a) the first, (b) the second and (c) the third parcel for the remote continental case. ($w_t$: liquid water content). The distributions are given per logarithmic radius interval.</figcaption></figure>

effect of collision and coalescence between the drops forms a second maximum in $g_w$ owing to the development of precipitation-sized drops. We notice that precipitation-sized drops do not develop until the third parcel (Fig. 6(c)). Collision and coalescence of drops are also responsible for the reduction of the second maximum in the drop number distribution function $f_d$ during the last stages of the ascent of the parcel (see Fig. 5(c)). In order to illustrate the development in time of precipitation-sized drops in the third parcel we have listed in Table 3(i) the number of drops per m³ larger than 30, 50 and 100 µm radius, at various times during the ascent of the third parcel. We notice that in the first parcel drops larger than 50 µm radius are present only in negligible numbers while in the third parcel drops large enough for precipitation are present after about 600 s, i.e. after an ascent of about 1700 m above cloud base. In Table 3(ii) we have displayed the effects of disregarding the collision-coalescence mechanism in our model
<table><caption>TABLE 3. NUMBER OF DROPS PER m³ WITH RADIi (a) LARGER THAN 30, 50, AND 100 µm IN THE THIRD AIR PARCEL AS A FUNCTION OF TIME FOR: (i) THE REMOTE CONTINENTAL CASE WITH COALESCENCEN CONSIDERED; (ii) THE REMOTE CONTINENTAL CASE WITHOUT CONSIDERING COALESCENCEN AND (iii) THE MARITIME CASE WITH COALESCENCEN CONSIDERED (* FOR THE TIME SINCE START OF FIRST THERMAL ADD 300 s)</caption><thead><tr><th>t (s)*</th><th>a > 30 µm</th><th>a > 50 µm</th><th>a > 100 µm</th></tr></thead><tbody><tr><td colspan="4">(i) remote continental (coalescence included)</td></tr><tr><td>400</td><td>1.6 × 10<sup>2</sup></td><td>0.3</td><td>3.6 × 10<sup>-8</sup></td></tr><tr><td>500</td><td>2.5 × 10<sup>3</sup></td><td>13</td><td>4.5 × 10<sup>-4</sup></td></tr><tr><td>600</td><td>3.7 × 10<sup>4</sup></td><td>5.6 × 10<sup>2</sup></td><td>0.5</td></tr><tr><td>700</td><td>1.6 × 10<sup>5</sup></td><td>9.8 × 10<sup>3</sup></td><td>34</td></tr><tr><td>800</td><td>3.7 × 10<sup>5</sup></td><td>4.9 × 10<sup>4</sup></td><td>1.0 × 10<sup>3</sup></td></tr><tr><td colspan="4">(ii) remote continental (coalescence not included)</td></tr><tr><td>400</td><td>2.9 × 10<sup>1</sup></td><td>5.1 × 10<sup>-4</sup></td><td>1.4 × 10<sup>-14</sup></td></tr><tr><td>500</td><td>1.8 × 10<sup>2</sup></td><td>3.1 × 10<sup>-3</sup></td><td>3.7 × 10<sup>-13</sup></td></tr><tr><td>600</td><td>1.7 × 10<sup>3</sup></td><td>2.1 × 10<sup>2</sup></td><td>6.3 × 10<sup>-12</sup></td></tr><tr><td>700</td><td>7.5 × 10<sup>3</sup></td><td>1.3 × 10<sup>-1</sup></td><td>4.9 × 10<sup>-11</sup></td></tr><tr><td>800</td><td>2.2 × 10<sup>4</sup></td><td>6.5 × 10<sup>-1</sup></td><td>2.8 × 10<sup>-10</sup></td></tr><tr><td colspan="4">(iii) maritime (coalescence included)</td></tr><tr><td>400</td><td>3.7 × 10<sup>4</sup></td><td>3.6 × 10<sup>2</sup></td><td>1.3 × 10<sup>-3</sup></td></tr><tr><td>500</td><td>2.5 × 10<sup>5</sup></td><td>9.9 × 10<sup>3</sup></td><td>7</td></tr><tr><td>600</td><td>8.9 × 10<sup>5</sup></td><td>1.0 × 10<sup>5</sup></td><td>1.6 × 10<sup>3</sup></td></tr><tr><td>700</td><td>1.2 × 10<sup>6</sup></td><td>2.5 × 10<sup>5</sup></td><td>2.2 × 10<sup>4</sup></td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Figure 7. Variation with time of the drop mass density distribution function g<sub>w</sub> for: (a) the first, (b) the second and (c) the third parcel for the maritime case. (w<sub>L</sub>: liquid water content). The distributions are given per logarithmic radius interval.</figcaption></figure>

calculations and have noted that the broadening of the drop size distribution is insufficient to produce precipitation-sized drops, if only condensation is considered.

(b) *Effect of successive convective cloud elements on the drop size spectrum—maritime case*

From Figs. 7(a, b, c) and Table 3(iii) we notice that drops larger than 50 µm radius develop much earlier (already after about 400 s) in the maritime case, than in the remote continental case. This can be seen already in Fig. 7(b) for the second parcel. The reason is that the maritime aerosol particle spectrum forms fewer drops. These, in turn, can
grow more efficiently, through condensation, beyond the size threshold at which the collision-coalescence process becomes effective. The ascent parameters $U$, $w_L$, and $s_{vw}$ for this case are not given here since they show similar behaviour to those given in Figs. 4(a, b and c). We notice, however, from Figs. 8(a, b and c) that the secondary maximum in the drop number distribution function $f_d$ develops much earlier and is more pronounced than in the remote continental case. Also, in this case, precipitation-sized drops develop in significant numbers only in the third parcel. We note, further, from Fig. 8(b) that a double maximum in $f_d$ develops at about 500 m below cloud top which is at 1600 m above cloud base, in good agreement with the observations for maritime clouds of Warner (1969, see his Fig. 5), given in Fig. 9. Also the positions of the maxima with respect to drop radius and the drop concentrations are in good agreement with Warner.

<figure><img src="image_9.png" /><figcaption>Figure 8. Variation with time of the drop number density distribution function $f_d$ for: (a) the first, (b) the second and (c) the third parcel for the maritime case. ($N_d$: total number of drops). The distributions are given per logarithmic radius interval.</figcaption></figure>

### (c) Effect of successive convective cloud elements on an inversion

The results illustrated in the previous section show that a succession of rising and condensing air parcels changes the thermodynamic characteristics of the cloud air. We therefore posed the question if it were possible for a succession of air parcels to penetrate through an inversion by altering the temperature structure typically associated with an inversion. For this purpose we assumed the vertical temperature and moisture distribution given in Fig. 10. In Fig. 11 we have illustrated the vertical distribution of the pertinent parcel parameters $U$, $w_L$, and $s_{vw}$ for the first (Fig. 11(a)), the second (Fig. 11(b)) and the third (Fig. 11(c)) air parcel. We notice from Figs. 11(a and b) that the rise of the first two air parcels is arrested by the inversion. However, the two successive air parcels have changed the vertical temperature and moisture profile. The effect of the air parcels' ascent on the environment is shown in Fig. 12 where the temperature and humidity profiles after the passage of the first, second and third parcels are compared with the initial profile. We notice that the third parcel is able to penetrate through the inversion (Fig. 11(c)) and that, in the parcel, drops of precipitable size have developed (Fig. 13).
<figure><img src="image_10.png" /><figcaption>Figure 9. Drop size distribution measured at the east coast of Australia in a convective cloud of 1500 m height about 300 m below cloud top by Warner (1969).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10. Initial vertical profile of temperature and humidity with inversion.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11. Variation with height and time of the vertical velocity *U*, the liquid water content *w*<sub>L</sub> and the supersaturation *S*<sub>vw</sub> for: (a) the first, (b) the second and (c) the third parcel for the remote continental aerosol particle spectrum for the vertical profile of temperature and humidity as given in Fig. 10.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12. Variation of the vertical temperature and humidity profile in the inversion layer: (a) initial profile as given in Fig. 10; (b) profile after passage of first parcel; (c) profile after passage of second parcel; (d) profile after passage of third parcel. For explanation see text.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 13. Variation with time of the drop mass distribution function $g_w$ for the third parcel for the remote continental aerosol particle spectrum with inversion. ($w_l$: liquid water content).</figcaption></figure>

4. CONCLUSIONS

The numerical evaluation of our multi-parcel cloud model shows that each rising air parcel establishes a new environment for the following air parcel, thus substantially affecting its thermodynamic behaviour and the microstructure of the cloud resulting from such preconditioning of the air.

Preconditioning of air by successive convective cloud elements has a significant effect on the evolution of the drop spectrum resulting from condensation, collision and coalescence inside a cloud, and, in particular on the time of appearance of precipitation-sized drops. Even if atmospheric temperature and humidity conditions are not favourable for precipitation-sized drops to appear, preconditioning of air by ascending air parcels may allow such drops to form. However, it is necessary to include the collision-coalescence mechanism in the calculations even during the early stages of a cloud, because by not considering this growth mechanism, not enough drops of radii larger than 50 µm will be produced, and it is these that initiate the formation of precipitation-sized drops. Air parcels rising successively in which drops grow by condensation and by collision-coalescence weaken an inversion and may eventually penetrate it.

A sensitivity study on the effects of the initial temperature excess of the parcel over its environment shows that our general conclusions are not affected by changing these initial conditions. This means that although the parcels may rise faster and reach higher levels in the atmosphere, our qualitative results, that each successive parcel further broadens the drop size spectrum and increases the probability of rain, still stand.
ACKNOWLEDGEMENTS

This research was funded by the Ministry for Research and Technology of the Federal Republic of Germany under project 325-4007-0704580. The authors of this paper are solely responsible for its content. The authors also acknowledge with gratitude the financial support of the German National Science Foundation.

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