# The role of droplet spectra for cloud radiative properties

Nikos Hatzianastassiou, Wolfram Wobrock, Andrea I Flossmann

► To cite this version:

Nikos Hatzianastassiou, Wolfram Wobrock, Andrea I Flossmann. The role of droplet spectra for cloud radiative properties. Quarterly Journal of the Royal Meteorological Society, 1997, 123 (544), pp.2215 - 2230. 10.1002/qj.49712354404 . hal-01905509

HAL Id: hal-01905509

https://uca.hal.science/hal-01905509v1

Submitted on 24 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

Q. J. R. Meteorol. Soc. (1997), 123, pp. 2215-2230

# The role of droplet spectra for cloud radiative properties

By NIKOS HATZIANASTASSIOU*, WOLFRAM WOBROCK and ANDREA I. FLOSMANN

Université Blaise Pascal, France

(Received 16 December 1996; revised 10 April 1997)

## SUMMARY

The evolution of different types of clouds is simulated with a spectral scavenging and microphysics model, DESCAM, coupled to the dynamics of an ascending and entraining air-parcel model. The resulting microphysical properties of the cloud are then coupled with a radiative-transfer model in order to calculate the optical properties, the up- and downgoing radiative fluxes, the cloud optical depth, and the cloud albedo. For a stratiform cloud investigated during the North Atlantic Regional Experiment good agreement with the experimental data is obtained. For a medium-sized convective cloud developing in marine or continental air the calculated cloud optical depth and the cloud albedo are compared with current parametrizations in global circulation models. The largest deviations are observed for cases with a large effective radius, e.g. when the cloud developed precipitation. In general, cloud optical depth and cloud albedo per unit cloud depth decrease with time, caused by the broadening of the cloud-drop spectrum.

**KEYWORDS:** Cloud modelling Optical properties Parametrization

## 1. INTRODUCTION

A large fraction of the earth's surface is covered by clouds. These clouds are currently understood to have an important impact on the earth's climate. They reflect parts of the short-wave radiation arriving from the sun, and they absorb and emit again long-wave terrestrial radiation. Presently, a global forcing of the climate due to the greenhouse gases is believed to be of the order of 2-3 W m⁻² (IPCC 1995). The capacity of aerosol particles to reflect solar radiation is currently estimated to yield a similar magnitude of the forcing but opposing the anticipated global warming (Kiehl and Briegleb 1993; IPCC 1995), whereby the indirect forcing due to clouds is estimated to be responsible for half of this value. However, due to the patchiness of the cloud fields this negative climate forcing is not counteracting the greenhouse warming evenly. Also, an anthropogenic change of the cloud-climate forcing is currently being discussed. For marine clouds, which represent a large fraction of the global cloud cover, it has been suggested that an enhanced dimethylsulfide production by the ocean might lead to an increase in concentration of small aerosol particles, increasing the number of small cloud drops formed and, thus, increasing cloud albedo (Charlson et al. 1978). From this it becomes clear that the representation of cloud formation, evolution and its radiative properties in global climate models is essential for a reliable simulation of current and future climate. Actually, the representation of clouds in climate models, however, is rather crude. Often they are described only by one or two liquid-water quantities, disregarding the fact that the size distribution of cloud droplets is important for correctly calculating the radiative cloud properties. For the moment these are simulated by assuming an effective cloud-drop radius (Slingo 1989, 1990; Slingo et al. 1989; Smith 1990; Boucher 1995).

The goal of this work is to compare current parametrizations for the cloud optical depth and cloud albedo in global models with detailed cloud microphysical and radiative simulations. Hereafter, we have used our spectral scavenging and microphysics model DESCAM coupled to the dynamics of a rising and entraining air parcel. The resulting drop size distributions have been used as input for the two-stream calculations in the radiation model of Zdunkowski et al. (1982). We have used a cloud microphysical and radiative data set for a stratus cloud obtained during the North Atlantic Regional Experiment (NARE) off

<footnote>* Corresponding author: Laboratoire de Météorologie Physique, Université Blaise Pascal - CNRS - OPGC, 24, avenue des Landais, F-63177 Aubière Cedex, France.</footnote>
the coast of Canada to verify our calculations against measurements. Encouraged by the
resulting good agreement we have applied our calculations to cumulus clouds developing
in two different air masses (marine and continental). These cumulus clouds have a higher
vertical extension and, consequently, allow one to study the microphysical changes and the
resulting consequences for the cloud radiative properties during a longer time of the cloud
parcel ascent. The cloud optical depth and the cloud albedo resulting from these detailed
calculations are then compared with the current parametrizations of these quantities in
global climate models.

In section 2 we will present the deployed dynamical, microphysical and radiative
models. The initial conditions for the stratus and the cumulus cloud simulations are de-
scribed in section 3 and in section 4 the results are discussed and compared with the global
model parametrizations.

2. MODEL DESCRIPTION

As discussed in detail by Flossmann *et al.* (1985) the basic framework employed in the present study is a rising and entraining air-parcel model coupled to DESCAM, i.e. the Detailed Scavenging and Microphysical model as discussed by Flossmann *et al.* (1985, 1987) and Flossmann and Pruppacher (1988). There, the aerosol particles are treated in a spectral form. Apart from dynamical processes, the number of particles of a certain size changes owing to activation to drops, size changes resulting from humidity changes, and impaction scavenging by drops. The nucleated drops then grow by condensation or evaporate, collide and coalesce, and eventually break up. During their lifetime they further scavenge particles and the scavenged pollutant mass is redistributed through the microphysical processes. The extension of this microphysical and scavenging model to the scavenging of two different types of aerosol particles (e.g. (NH₄)₂SO₄ and NaCl for marine air masses) is described by Flossmann (1991) and the inclusion of effects of gaseous absorption (H₂O₂ and O₃) on the uptake and oxidation of SO₂ was presented by Flossmann (1994). Currently, the model contains prognostic equations for 11 distribution functions: the drop number density distribution function f₉(m), the mass density distribution function gAPd,(NH₄)₂SO₄(m) and gAPd,NaCl(m) for (NH₄)₂SO₄ and NaCl particles in the drops, respectively. In addition the mass density distribution functions gGd,S(4)(m), gGd,S(6)(m), gGd,H₂O₂(m), and gGd,O₃(m), for the sulphur species S(4) and S(6), and H₂O₂ and O₃ gas in the drops, respectively, are predicted, with *m* being the drop mass discretized in 69 radius size bins. The aerosol particle number density distribution functions fAPa,(NH₄)₂SO₄(mAP) and fAPa,NaCl(mAP) for (NH₄)₂SO₄ and NaCl particles in the air, and the mass density distribution functions gAPa,(NH₄)₂SO₄(mAP) and gAPa,NaCl(mAP) for (NH₄)₂SO₄ and NaCl particles in the air, with *m*AP being the aerosol particle mass discretized in 81 radius size bins are also predicted. For further details on the terms changing the density distribution functions see Flossmann *et al.* (1985) and Flossmann (1994).

The radiation model used is described by Zdunkowski *et al.* (1982). It treats the
multiple scattering with a practical improved flux method (Zdunkowski *et al.* 1980) for the
solar spectrum and the atmospheric window region. For the remaining infrared spectrum an
extended emissivity method is used, taking into account all the overlapping effects and grey
clouds. The accuracy of this method is discussed by Zdunkowski and Breslin (1979). The
model incorporates the effects of atmospheric water vapour (H₂O), carbon dioxide (CO₂),
ozone (O₃), pollution gases such as NO₂, and dry air and aerosol particles. The spectral
integration is made in six spectral intervals covering the short-wave (SW) and infrared (IR)
regions. The optical properties of atmospheric gases permit the splitting of the IR emission
spectrum into two major spectral intervals: in the window region (8.75–12.25 µm) the
gaseous absorption is relatively weak so that scattering effects by particles are considered. In the remaining IR spectral interval (3.7–8.75 and 12.25–100 µm) scattering effects are neglected owing to the dominant gas and droplet absorption. The complexity of the solar absorption of atmospheric gases as well as the significant wavelength dependence of droplet absorption require the subdivision of the SW spectrum into four intervals: (0.28–1.0 µm), (1.0–1.53 µm), (1.53–2.2 µm) and (2.2–6.0 µm). The model requires the extinction and absorption coefficients ($\sigma_{ext}$, $\sigma_{abs}$) as well as the asymmetry factor $g$ of the phase function for the water droplet and the aerosol particle spectra. These quantities were originally parametrized for various types of clouds (cumulus, stratus and altostratus) and aerosol particle spectra (urban, rural, maritime). Our contribution was to substitute these parametrized optical properties by the ones which we calculated through Mie theory for the cloud constituents resulting from the air-parcel model.

To provide the above-mentioned optical properties a Mie calculation code was formulated which retrieves the extinction, absorption and scattering coefficients ($\sigma_{ext}$, $\sigma_{abs}$ and $\sigma_{scat}$), the single-scattering albedo ($\omega$), the asymmetry factor $g$ and the phase function of the scattering angle for the droplet and aerosol spectra considered. The composition of the cloud particles is taken into account through the calculation of the refractive indices (real and imaginary) of the individual components of each cloud constituent. The refractive indices for water stemmed from Hale and Querry (1973) while the ones for the water soluble and non-soluble materials were compiled from Shettle and Fenn (1976) and are tabulated in d'Almeida et al. (1991). The effect of the scavenged aerosol particles on the refractive indices of the water droplets was considered following Hanel and Lehman (1981). The Mie computations were performed for stepwidths ($\log(r_{i+1}/r_i)$) equal to 0.05, where the index $i$ refers to the spectral category of the particle radius $r$.

### 3. INITIAL CONDITIONS

#### (a) For the stratus case

NARE, an IGAC*-sanctioned experiment, took place between 15 August and 8 September 1993. The measurements for the chosen data set were conducted over the Bay of Fundy and the Gulf of Maine within a 100 km radius of Yarmouth, Nova Scotia, Canada (near 44°N, 66°W).

Measurements were made from the National Research Council of Canada DH-6 Twin Otter aircraft and at a coastal ground site (Chebogue Point) 10 km south of Yarmouth. The data set was made available to us through the 4th International Cloud Modelling Workshop in Clermont-Ferrand, 12–16 August 1996. The NARE results and data set are published in a special issue of the *Journal of Geophysical Research*. For a description of the dynamical and microphysical measurements see, for example, Leaitch et al. (1996). The aerosol measurements are described by Liu et al. (1996) and Li et al. (1996).

From the data set we chose 1500 UTC 7 September 1993. The measured temperature and humidity profile yielded the input sounding for the model simulations. To obtain the initial aerosol particle spectrum for model input we took the aerosol size distribution measured at Chebogue Point outside the cloud and compared it with a fit considering three log-normal distributions:

$$dN_{APa}/d \log r_N = \sum_{i=1}^{3} \frac{N_i}{\sqrt{2\pi} \log(\sigma_i)} \exp \left( - \frac{\{\log(r_N/R_i)\}^2}{2(\log(\sigma_i))^2} \right) \quad (1)$$

<footnote>* International Global Atmospheric Chemistry study.</footnote>
<table><caption>TABLE 1. PARAMETERS OF THE THREE LOG-NORMAL MODES OF THE AEROSOL PARTICLE DISTRIBUTION IN EQ. (1) AT CHEBOGUE POINT FOR THE STRATUS CASE</caption><thead><tr><th>Mode</th><th>Number N<sub>i</sub> (cm<sup>-3</sup>)</th><th>Geometric radius R<sub>i</sub> (μm)</th><th>Standard deviation logσ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>1833</td><td>0.039</td><td>0.457</td></tr><tr><td>2</td><td>170.6</td><td>0.113</td><td>0.1</td></tr><tr><td>3</td><td>0</td><td>0.29</td><td>0.396</td></tr></tbody></table>

where $r_N$ is the radius of the 'dry' aerosol particle, $N_{APa}$ is the total particle number concentration (in cm$^{-3}$), $N_i$ is the total number of aerosol particles (per cm$^3$), $R_i$ is the mean particle radius (in μm) and $\sigma_i$ is the standard deviation of the i-th log-normal distribution.

The values for the parameters are given in Table 1, and the measured spectrum and the derived model input are shown in Fig. 1(a). Simultaneous measurements of the aerosol particle chemical composition indicate among other components a strong sulphate contribution. Consequently, we assumed all particles to consist of 100% $(NH_4)_2SO_4$. The resulting cloud condensation nucleus (CCN) spectrum is compared with the observed one in Fig. 1(b).

The model parcel was launched at 985 hPa altitude starting with a relative humidity of 99% and with a vertical velocity of 0.4 m s$^{-1}$. The model time step was set to 2 s with a reduction in the condensation routine whenever necessary. The resulting cloud droplet and aerosol spectra were stored every 20 s to yield the input conditions for the radiation model.

The radiation model was initialized apart from the cloud droplet with an aerosol particle spectrum and trace gases (CO$_2$, NO$_2$, O$_3$) inside and outside the cloud region. The aerosol particle spectra inside the cloud were the ones calculated from the microphysical model at the corresponding height. The vertical aerosol particle profiles outside cloud were taken from the measurements in the maritime boundary layer carried out during the ascent, flight 45 at 1500 UTC 7 September 1993. The O$_3$ distribution was taken from Craig (1965) and a constant CO$_2$ concentration with height was used, as in Zdunkowski *et al.* (1982). The pollution gas NO$_2$ was taken as 0.2 parts per million at the surface and decreasing linearly within the first 1000 m, and zero concentration above. The surface temperature was set to 288.6 K. Furthermore, the measured temperature and relative-humidity profiles yielded the input atmospheric conditions for the radiation model. The surface emissivity was set to 0.98 (the mean emissivity value for water and ocean surfaces). The ground albedo over the water was set to 0.1. The radiation code was initialized for 44° latitude and for the Julian date 250 (7 September) at 1500 UTC. The grid spacing of the model changed. It was equal to 25 m between the surface and 650 m altitude (including the cloud region from 300 m to 600 m) and it increased above for the remaining levels, stretching from 820 m to 59 km.

### (b) For the cumulus case

The initial vertical profile for temperature and humidity for the medium-sized cumulus cloud was the one used already by Lee *et al.* (1980). The initial dry aerosol particle spectrum was again given by Eq. (1). The parameters we used pertain to a typical marine and a typical continental air mass taken from Jaenicke (1988) and are summarized in Tables 2 and 3. All particles were again assumed to consist of 100% $(NH_4)_2SO_4$.
<figure><img src="image_3.png" /><figcaption>Figure 1. (a) Initial aerosol particle spectrum resulting from Eq. (1) and Table 1 (solid line) and the measured aerosol particle spectrum at Chebogue Point on 7 September 1993 at 1500 UTC (dashed line). (b) Initial cloud condensation nucleus (CCN) spectrum corresponding to the solid line of Fig. 1(a) (solid line) and the measured CCN spectrum (dashed line).</figcaption></figure>

<table><caption>TABLE 2. PARAMETERS OF THE THREE LOG-NORMAL MODES FOR THE CONTINENTAL AEROSOL PARTICLE DISTRIBUTION IN EQ. (1) FOR THE CUMULUS CASE</caption><thead><tr><th>Mode</th><th>Number<br>N<sub>i</sub> (cm<sup>-3</sup>)</th><th>Geometric radius<br>R<sub>i</sub> (μm)</th><th>Standard deviation<br>log σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>6650</td><td>0.00739</td><td>0.225</td></tr><tr><td>2</td><td>147</td><td>0.0269</td><td>0.557</td></tr><tr><td>3</td><td>1990</td><td>0.04190</td><td>0.266</td></tr></tbody></table>

From Jaenicke (1988).
<table><caption>TABLE 3. PARAMETERS OF THE THREE LOG-NORMAL MODES FOR THE MARITIME AEROSOL PARTICLE DISTRIBUTION IN EQ. (1) FOR THE CUMULUS CASE</caption><thead><tr><th>Mode</th><th>Number N<sub>i</sub> (cm<sup>-3</sup>)</th><th>Geometric radius R<sub>i</sub> (μm)</th><th>Standard deviation logσ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>133</td><td>0.0039</td><td>0.657</td></tr><tr><td>2</td><td>66.6</td><td>0.133</td><td>0.21</td></tr><tr><td>3</td><td>3.06</td><td>0.29</td><td>0.396</td></tr></tbody></table>

From Jaenicke (1988).

The model parcel was launched at 900 hPa with an initial relative humidity of 99% and a vertical velocity of 1 m s⁻¹. The time step used was calculated as in the stratus case. The droplet spectra were stored every 100 s and the cloud region extended from 1000 m to 4108 m for the maritime case while for the continental one it stretched between 1000 m and 4322 m.

The radiation model was initialized as in the stratus case. The vertical profiles of temperature and relative humidity were the same as for the air-parcel model. The grid distance was 50 m for the first 1000 m for both rural and maritime cases. In the cloud region the grid distance changed so that the radiation-code levels correspond to the altitudes where the droplet spectra were stored. Above the cloud region and up to 59 km the grid distance increased for both cases, continental and maritime. The ground emissivity and ground albedo were assumed to have typical values for agricultural ground and water surface for the continental and maritime cloud, respectively. Thus, the ground emissivity was 0.98 and 0.92, and the ground albedo had values of 0.1 and 0.2, for the maritime and the continental cases, respectively. The cloud droplet spectra were the ones derived from the air-parcel model for the two types of cloud. From the simulation of the stratus cloud we noted that the effect of the presence of interstitial aerosol particles on the cloud albedo is small and is dominated by the cloud variability. Consequently, interstitial aerosol particles were disregarded for the simulations of the radiative properties of the cumulus cloud. A constant CO₂ concentration with height was used (Zdunkowski *et al.* 1982) and the O₃ distribution was again taken from Craig (1965). The NO₂ distribution was the same as for the stratus case. All the simulations were performed for 1200 UTC 30 July (Julian date 212) and for a latitude of 50°.

## 4. MODEL RESULTS

### (a) For the stratus case

When the air parcel was launched at 300 m it immediately reached supersaturation and a cloud formed. The parcel rose inside the cloud and stopped at the cloud top which was at 600 m. Figure 2 shows the evolution with time, i.e. height, of the liquid-water content and the total number concentrations of drops resulting from the model simulation, and compares it with the available observations. We note, here, that the fluctuations in the measurements cannot be reproduced by an ascending air-parcel model using a homogeneous entrainment approach. Here, three-dimensional large-eddy simulation models would be the appropriate tool to reproduce these fluctuations. However, our goal is not the study of the turbulence structure of the stratus cloud but to obtain model droplet spectra which are representative for the cloud in question. Figure 3 shows a comparison of the modelled and observed drop size distribution at 576 m altitude (near the cloud top). The measured spectrum shows a slightly higher number of small drops and a lower number of large drops. This might be
<figure><img src="image_4.png" /><figcaption>Figure 2. Evolution with height of the liquid-water content (LWC) and the droplet total number concentration ($N_d$) and comparisons with observations.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3. Drop size distribution near cloud top. The solid line is the drop spectrum derived from the model at 576 m. The dashed line is the observed drop spectrum at 576 m (near cloud top).</figcaption></figure>

improved by a better representation of entrainment. In general, however, both figures show a good agreement between the simulation and the observations.

This agreement encouraged us to derive the radiation upward and downward fluxes. Therefore, we first derived the optical parameters from Mie theory. Figure 4 shows $\omega$ and g for the model droplet size distribution of Fig. 3. The single-scattering albedo follows the maxima and minima in the real refractive index for water with three large minima at 3, 6 and 10 µm (Liou 1992). For wavelengths larger than 10 µm the size-parameter effect becomes more important. The size parameter $\alpha$ is defined as the ratio of particle circumference to incident wavelength ($\alpha = 2\pi r/\lambda$). The asymmetry factor remains about the same with values of about 0.8-0.85 for wavelengths up to about 10 µm, with the exceptions of 3
<figure><img src="image_6.png" /><figcaption>Figure 4. Single-scattering albedo, $\omega$, and asymmetry factor, $g$, for the model droplet spectra of Fig. 3 at 576 m altitude versus wavelength, $\lambda$. The albedo and the asymmetry factor are dimensionless numbers.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. Up- and downwelling short-wave fluxes as functions of height.</figcaption></figure>

and 6 µm; at these wavelengths strong forward scattering occurs due to diffraction. For wavelengths larger than 10 µm both $\omega$ and $g$ drastically decrease owing to the reduction of $\alpha$. Figure 5 shows the resulting up- and downwelling fluxes resulting from the radiation model. Here, we can see a reasonable agreement with the measurements made. We see significant deviations only above cloud top and near the surface for the downward flux. The deviations near the surface can be attributed to the fact that the two-stream model is not able to reproduce well fluxes above the ground because it does not carry enough information about the lower-boundary conditions. Another explanation for the deviation could be the fact that the cloud was thinner when the measurements were made near the surface. The deviations above cloud top may be due to horizontal inhomogeneities in the stratus cloud top which are not considered presently by the model. Obviously, the fact that our model cannot treat the natural inhomogeneities in clouds inhibits a more thorough
<figure><img src="image_8.png" /><figcaption>Figure 6. Size distribution of the liquid-water mass $g_w$: (a) for the continental air mass and (b) for the maritime air mass at the beginning (200 s) and the end (1900 s) of the cloud cycle.</figcaption></figure>

reproduction of the measurements. In using the observed droplet spectrum displayed in Fig. 3 we can, however, estimate that this would result only in an increase of the upwelling flux of the order of about 0.3%.

Calculating the cloud optical depth and the cloud albedo from the fluxes we obtain values of 21.4 and 0.7, respectively. Unfortunately, these values cannot be compared with measurements as no satellite measurements were included in the NARE campaign.

Considering the results obtained and their good agreement with the observations we felt confident in extending the application of our model simulations to cumulus clouds forming in different air masses.

(b) For the cumulus case

For the cumulus case we investigated the cloud optical thickness and the cloud albedo for clouds forming on two different aerosol particle spectra as given by Eq. (1) and Tables 2 and 3. During the rise of the air parcel the droplet spectrum evolved due to the microphysical processes. Figure 6 shows the spectral distribution of the liquid-water mass for the two air
<table><caption>TABLE 4. CONTINENTAL CLOUD OPTICAL DEPTH AND ALBEDO RESULTING FROM THE DE-TAILED MODEL RESULTS</caption><thead><tr><th colspan="4">(a)</th></tr><tr><th colspan="4">Cloud optical depth</th></tr><tr><th></th><th>At 200 s</th><th>At 1900 s</th><th>At 1900 s<br>(normalized)</th></tr></thead><tbody><tr><th>τ<sub>1</sub></th><td>27.4</td><td>364.9</td><td>25.7</td></tr><tr><th>τ<sub>2</sub></th><td>25.2</td><td>351.5</td><td>24.8</td></tr><tr><th>τ<sub>3</sub></th><td>27.8</td><td>1544.3</td><td>108.8</td></tr><tr><th>τ<sub>4</sub></th><td>26.0</td><td>575.6</td><td>40.5</td></tr><tr><th>τ<sub>5</sub></th><td>25.2</td><td>351.6</td><td>24.8</td></tr></tbody></table>

(b)

<table><thead><tr><th colspan="4">Cloud albedo</th></tr><tr><th></th><th>At 200 s</th><th>At 1900 s</th><th>At 1900 s<br>(normalized)</th></tr></thead><tbody><tr><th>A</th><td>0.76</td><td>0.97</td><td>0.76</td></tr><tr><th>Aτ<sub>1</sub></th><td>0.82</td><td>0.98</td><td>0.82</td></tr><tr><th>Aτ<sub>2</sub></th><td>0.81</td><td>0.98</td><td>0.82</td></tr><tr><th>Aτ<sub>3</sub></th><td>0.82</td><td>0.99</td><td>0.95</td></tr><tr><th>Aτ<sub>4</sub></th><td>0.81</td><td>0.99</td><td>0.88</td></tr><tr><th>Aτ<sub>5</sub></th><td>0.81</td><td>0.98</td><td>0.82</td></tr></tbody></table>

τ₁ and different parametrizations τ₂ to τ₅ as given by Eqs. (4), (6), (8) and (9) for a continental cloud at the beginning and the end of its life cycle; in the right-hand column the results for 1900 s are normalized to the same cloud thickness as at 200 s.

masses considered at two different times, one pertaining to the beginning of the simulation and the other to the end. We see here the different microphysical evolution of the cloud for the continental air mass as compared with the maritime. In the continental cloud (Fig. 6(a)) 1099 cm⁻³ aerosol particles served as CCNs forming the same number of drops. Having to share the supply of water vapour the drops grew slowly beyond the size where collision and coalescence become effective. This is in contrast to the maritime aerosol particle spectrum; it produced 89 cm⁻³ CCNs. This lower drop-number concentration resulted in a stronger growth of the drops. Collision and coalescence then rapidly formed precipitation-sized drops as can be seen in Fig. 6(b).

The radiation model yielded the cloud optical thickness and the cloud albedo which are given in Tables 4 and 5. We assumed, here, that the entire cloud is made of the cloud drop spectrum at 200 s or 1900 s, respectively. We are aware of the fact that a cloud has a heterogeneous structure. We have refrained, however, from attributing the spectra obtained for the different model times to the corresponding altitude as we have done for the stratus case. This was done for two reasons. The first is that for the stratus case the cloud only extended over a height of 300 m. The parcel covered this distance in 7 min where not much evolution of the drop spectrum can be anticipated. This is different for a cumulus cloud extending at the end over about 3 km where it took the parcel 30 min to cover this distance. Here, we cannot safely assume that the microphysical properties at
<table><caption>TABLE 5. MARITIME CLOUD OPTICAL DEPTH AND ALBEDO RESULTING FROM THE DETAILED MODEL RESULTS</caption><thead><tr><th rowspan="2"></th><th colspan="3">Cloud optical depth</th></tr><tr><th>At 200 s</th><th>At 1900 s</th><th>At 1900 s<br>(normalized)</th></tr></thead><tbody><tr><th>τ<sub>1</sub></th><td>11.0</td><td>17.1</td><td>1.2</td></tr><tr><th>τ<sub>2</sub></th><td>10.6</td><td>18.4</td><td>1.3</td></tr><tr><th>τ<sub>3</sub></th><td>12.6</td><td>4749.8</td><td>346.9</td></tr><tr><th>τ<sub>4</sub></th><td>11.2</td><td>117.2</td><td>8.6</td></tr><tr><th>τ<sub>5</sub></th><td>10.6</td><td>18.4</td><td>1.3</td></tr></tbody></table>

<table><thead><tr><th rowspan="2"></th><th colspan="3">Cloud albedo</th></tr><tr><th>At 200 s</th><th>At 1900 s</th><th>At 1900 s<br>(normalized)</th></tr></thead><tbody><tr><th>A</th><td>0.56</td><td>0.57</td><td>0.19</td></tr><tr><th>Aτ<sub>1</sub></th><td>0.65</td><td>0.77</td><td>0.2</td></tr><tr><th>Aτ<sub>2</sub></th><td>0.64</td><td>0.79</td><td>0.21</td></tr><tr><th>Aτ<sub>3</sub></th><td>0.68</td><td>0.99</td><td>0.99</td></tr><tr><th>Aτ<sub>4</sub></th><td>0.66</td><td>0.96</td><td>0.63</td></tr><tr><th>Aτ<sub>5</sub></th><td>0.64</td><td>0.79</td><td>0.21</td></tr></tbody></table>

$τ_1$ and different parametrizations $τ_2$ to $τ_5$ as given by Eqs. (4), (6), (8) and (9) for a maritime cloud at the beginning and the end of its life cycle; in the right-hand column the results for 1900 s are normalized to the same cloud thickness as at 200 s.

the different heights persisted unchanged during the cloud evolution. The second reason is that parametrizations of cloud optical depth and cloud albedo in global models also mostly do not contain a time or height evolution of the cloud. Consequently, we chose to look at clouds homogeneously, consisting of a drop spectrum representing the early stage of its evolution where the cloud has a height of 234 m and 227 m for the continental and the maritime case, respectively, and one representing the later stage where the cloud has a height of 3322 m and 3108 m for the continental and the maritime case, respectively, and compare these results with parametrizations also assuming a homogeneous cloud.

Thus, also displayed in Tables 4 and 5 is a comparison of our detailed model results with some parametrizations of cloud optical depth and cloud albedo as currently used in global climate modelling.

The formulae are summarized below:

(i) For the cloud optical depth. The optical depth for a given droplet size distribution is defined by (see Stephens 1978):

$$τ_1 = \pi \int_0^{\Delta z} \int_{r_1}^{r_2} Q_{\text{ext}} r^2 f(r) \, dr \, dz \quad (2)$$
which for our assumption of homogeneity becomes:

$$ \tau_1 = \pi \Delta z \int_{r_1}^{r_2} Q_{\text{ext}} r^2 f(r) \, dr \quad (3) $$

where: $\Delta z$ is the vertical extension of the cloud, $Q_{\text{ext}}$ is the extinction efficiency, $r$ is the radius of the droplet, $f(r)$ is the droplet number distribution function, and $r_1$ and $r_2$ are the lower and upper limits for the droplet radius range. For narrow droplet-size distributions all droplets are regarded as having the same radius, whereby the extinction efficiency, which is a function of the droplet radius, wavelength and refractive index, can be considered to have a fixed value. This value of $Q_{\text{ext}}$ for the solar visible wavelengths is about 2 (Liou 1992) and consequently Eq. (3) becomes (Arends 1996):

$$ \tau_2 = 2\pi \Delta z \int_{r_1}^{r_2} r^2 f(r) \, dr. \quad (4) $$

Using the mean effective radius which is defined (Hansen and Travis 1974) as:

$$ r_{\text{eff}} = \frac{\int_{r_1}^{r_2} \pi r^3 f(r) \, dr}{\int_{r_1}^{r_2} \pi r^2 f(r) \, dr} \quad (5) $$

and replacing the integrated squared radius over the droplet size distribution in Eq. (4) with the square of the mean effective radius we can approximate (Schwartz and Slingo 1996):

$$ \tau_3 = 2\pi \Delta z N r_{\text{eff}}^2 \quad (6) $$

where $N$ is the total droplet number. Another possibility is to replace the integrated squared radius over the droplet size distribution with the mean water-volume radius which is defined (Platnick and Twomey 1994) as:

$$ r_{\text{volw}}^3 = \frac{\int_{r_1}^{r_2} r^3 f(r) \, dr}{\int_{r_1}^{r_2} f(r) \, dr} \quad (7) $$

leading to the volume-weighted optical depth (King *et al.* 1995)

$$ \tau_4 = 2\pi \Delta z N r_{\text{volw}}^2. \quad (8) $$

Finally we used the formula proposed by Stephens (1978) which introduces the liquid-water path (LWP) and the effective radius of the cloud:

$$ \tau_5 = \frac{3 \text{ LWP}}{2 r_{\text{eff}}}. \quad (9) $$

LWP is the vertically integrated liquid-water content (Liou 1992):

$$ \text{LWP} = \Delta z \frac{4\pi}{3} \rho_1 \int_{r_1}^{r_2} r^3 f(r) \, dr \quad (10) $$

where $\rho_1$ is density of the water.

(ii) *For the cloud albedo.* The following formula, proposed by Meador and Weaver (1980), is considered to be a good approximation for the cloud albedo, A (Hobbs 1993):

$$ A = \frac{(1 - g)\tau}{1 + (1 - g)\tau}. \quad (11) $$
This formula contains the asymmetry factor 'g' calculated from Mie theory and the optical depth $\tau$ of the cloud. For $\tau$ we can use either the value $\tau_1$ which is the detailed simulated value or the parametrized values $\tau_2-\tau_5$ to obtain the albedo values $A_{\tau1}, A_{\tau2}, A_{\tau3}, A_{\tau4}$ and $A_{\tau5}$ which are compared with the value A calculated directly from the two-stream flux model.

From Table 4(a) for the continental case we see that the values for the optical thickness at 200 s are very close to each other while at 1900 s the values for $\tau_3$ and $\tau_4$ differ significantly from the rest. This is due to the fact that at 1900 s the production of (even) a few big droplets from the cloud led to the increase of (mainly) the mean effective radius and also the mean water-volume radius whose squared values account for the optical depth $\tau$ (Eqs. (5) and (7)). Actually at 1900 s they had values of $r_{\text{eff}} = 22.31$ µm and $r_{\text{volw}} = 13.62$ µm, respectively.

For the cloud albedo A in Table 4(b) we also see that for 200 s all parametrized values agree very closely. For 200 s as well as 1900 s we note that A is overestimated when calculated using Eq. (11). This overestimation for the continental case is smaller for 1900 s because of the high values of $\tau(> 350)$ leading to values of A around 1, but it is more evident in Table 5(b) for the maritime cloud.

In the right-hand column of Table 4(a) the results for 1900 s are normalized to the cloud thickness at 200 s (234 m for the continental case). The difference between $\tau_3$ and $\tau_4$ and the other cloud optical depths ($\tau_1, \tau_2$ and $\tau_5$) at 1900 s is reduced due to the fact that the cloud depth at 1900 s is smaller now. We see, here, the dependence of $\tau$ on the cloud droplet spectra, as at 1900 s $\tau$ has slightly decreased due to the small broadening of the droplet spectra. For the normalized values A does not change substantially from 200 s to 1900 s, because no large broadening of the cloud droplet spectra has taken place. Nevertheless, we see an overestimation in A through Eq. (11) and using Eqs. (6) and (8) with $r_{\text{eff}}$ and $r_{\text{volw}}$, respectively.

For the maritime cloud displayed in Table 5 this behaviour is even more pronounced due to the more rapid development of precipitation-sized drops in the droplet spectrum. We note in general that Eq. (11), using the parametrization formulae (4) and (9), gives results which are not much different from those when using Eq. (3). In the right-hand column of Table 5 the cloud thickness at 1900 s is constant and equal to 227 m (i.e. the value it had at 200 s for the maritime case). The small increase of $\tau$ at 1900 s is now replaced by an important decrease, i.e. the effect of the strong broadening of the cloud droplet spectra which produces a reduction of $\tau$ is now visible and not counterbalanced by the growth in cloud depth. As a result, instead of the increase of A at 1900 s, a significant decrease of A can be seen, leading to values near 0.20 except for the parametrized albedo calculated via Eq. (11) using Eqs. (6) and (8), which contain the high values $r_{\text{eff}} = 385.11$ µm and $r_{\text{volw}} = 60.49$ µm, respectively.

Consequently, we can conclude that for the cases of precipitation development the formulae depending on the cloud effective radius or the mean volume radius largely over-predict the optical depth and the cloud albedo. This statement is in agreement with Schwartz and Slingo (1996) who stated that for the calculation of the optical depth for broader droplet distributions it is necessary to use the integrated squared radius over the droplet size distributions as done in Eq. (4).

## 5. SUMMARY AND CONCLUSIONS

We have simulated the evolution of different clouds with our DESCAM model coupled to the framework of a rising and entraining air parcel. The resulting microphysical drop
spectra then entered a radiation code to yield the up- and downwelling radiative fluxes, the cloud optical depth and the cloud albedo.

For a stratiform cloud we have taken a data set obtained during the NARE campaign. Initializing the model with measurements of the aerosol particle distribution and chemical composition yielded microphysical and radiative properties which agreed well with the available observations. This encouraged us to use this approach to study the radiation parametrizations currently used in global climate models. We did so for a small convective cloud forming in two different air masses at two different times during cloud evolution. We found, in general, that the cloud optical depth and the cloud albedo per unit cloud depths decreased with time owing to the anticipated broadening of the drop spectrum owing to cloud microphysical processes. This was especially pronounced for the case of the maritime cloud where precipitation-sized drops were rapidly formed. In the continental cumulus, where fewer precipitation drops formed (because of the fact that collision and coalescence were less effective), the cloud albedo and optical thickness changed only slightly during the cloud lifetime cycle. The parametrizations generally agreed to within 20% with the detailed simulations, whereby the largest error is connected to the parametrization including the effective radius or mean volume radius, and for the case that precipitation-sized drops developed. For the future it is planned to extend our studies from the air-parcel framework to a two-dimensional framework, where the evolution of the drop microphysics can be more realistically modelled as a function of time and height and, consequently, also the evolution of the radiative properties of the cloud. Furthermore, it is planned to test the model with the data that will be obtained during the ACE2/CLOUDYCOLUMN experiment planned for July 1997 around the Canary Islands (Spain) which will, then, also allow the comparison with cloud optical measurements and cloud albedos observed *in situ* and via satellite.

## ACKNOWLEDGEMENTS

The authors are solely responsible for the content of this manuscript and acknowledge with gratitude the support by the European Community under ENV4-CT95-0117 and ENV4-CT95-0012. The calculations for this paper were done on the CRAY C98 and C94 of the 'Institut du Développement et des Resources en Informatique Scientifique' in Orsay (France) under project No. 940180. The authors acknowledge with gratitude the hours of computer time and the support provided. Furthermore, the authors acknowledge the support provided by the French national program PATOM (Programme Atmosphère et Océan à Moyenne Échelle).

## REFERENCES

<table><tbody><tr><td>Almeida, G. A., Koepke, P. and Shettle, E. P.</td><td>1991</td><td>Atmospheric aerosols. Pp. 57–59 in <i>Global climatology and radiative characteristics</i>. A. Deepak publishing</td></tr><tr><td>Arends, B. G.</td><td>1996</td><td>'Aerosol and cloud microphysics, measurements and interpretations'. Ph.D. thesis, University of Utrecht</td></tr><tr><td>Boucher, O.</td><td>1995</td><td>GCM estimate of the indirect aerosol forcing using satellite-retrieved cloud effective droplet radii. <i>J. Climate</i>, 8, 1403–1409</td></tr><tr><td>Charlson, R. J., Lovelock, J. E., Andreae, M. O. and Warren, S. G.</td><td>1987</td><td>Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate. <i>Nature</i>, 326, 655–661</td></tr><tr><td>Craig, R. A.</td><td>1965</td><td><i>The upper atmosphere, meteorology and physics</i>. Academic Press, New York and London</td></tr><tr><td>Flossmann, A. I.</td><td>1991</td><td>The scavenging of two different types of marine aerosol particles using a two-dimensional detailed cloud model. <i>Tellus</i>, 43B, 301–321</td></tr></tbody></table>
<table>
  <tr>
    <td>Flossmann, A. I.</td>
    <td>1994</td>
    <td>A 2-D spectral model simulation of the scavenging of gaseous and particulate sulfate by a warm marine cloud. J. Atmos. Res., 32, 255-268</td>
  </tr>
  <tr>
    <td>Flossmann, A. I. and Pruppacher, H. R.</td>
    <td>1988</td>
    <td>A theoretical study of the wet removal of atmospheric pollutants. Part III. J. Atmos. Sci., 45, 1857-1871</td>
  </tr>
  <tr>
    <td>Flossmann, A. I., Hall, W. D. and Pruppacher, H. R.</td>
    <td>1985</td>
    <td>A theoretical study of the wet removal of atmospheric pollutants. Part I. J. Atmos. Sci., 42, 582-606</td>
  </tr>
  <tr>
    <td>Flossmann, A. I., Pruppacher, H. R. and Topalian, J. H.</td>
    <td>1987</td>
    <td>A theoretical study of the wet removal of atmospheric pollutants. Part II. J. Atmos. Sci., 44, 2912-2923</td>
  </tr>
  <tr>
    <td>Hale, G. M. and Querry, M. R.</td>
    <td>1973</td>
    <td>Optical constants of water in the 200 nm to 200 µm wavelength region. Appl. Opt., 12, 555-563</td>
  </tr>
  <tr>
    <td>Hanel, G. and Lehman, M.</td>
    <td>1981</td>
    <td>Equilibrium size of aerosol particles and relative humidity: New experimental data from various aerosol types and their treatment for cloud physics application. Contrib. Atmos. Phys., 54, 57-71</td>
  </tr>
  <tr>
    <td>Hansen, J. E. and Travis, L. D.</td>
    <td>1974</td>
    <td>Light scattering in planetary atmospheres. Space Sci. Rev., 16, 527-610</td>
  </tr>
  <tr>
    <td>Hobbs, P. V.</td>
    <td>1993</td>
    <td>Aerosol-cloud-climate interactions. International Geophysics Series. Academic Press Inc.</td>
  </tr>
  <tr>
    <td>IPCC</td>
    <td>1995</td>
    <td>Climate change 1995, The science of climate change. Eds. J. T. Houghton, L. G. Meira Filho, B. A. Callander, N. Harris, A. Kattenberg and K. Maskell. Cambridge University Press</td>
  </tr>
  <tr>
    <td>Jaenicke, R.</td>
    <td>1988</td>
    <td>Aerosol physics and chemistry. Pp. 391-457 in Landolt-Boernstein: Zahlenwerte und Funktionen aus Naturwissenschaften und Tecknik, V 4b. Ed. G. Fisher. Springer</td>
  </tr>
  <tr>
    <td>Kiehl, J. T. and Briegleb, B. P.</td>
    <td>1993</td>
    <td>The relative roles of sulfate aerosols and greenhouse gases in climate forcing. Science, 260, 311-314</td>
  </tr>
  <tr>
    <td>King, M. D., Tsay, S.-C. and Platnick, S.</td>
    <td>1995</td>
    <td>In situ observations of the indirect effects of aerosols on clouds. In Aerosol forcing of climate. Eds. R. J. Charlson and G. Heintzenberg. John Wiley and Sons Ltd</td>
  </tr>
  <tr>
    <td>Leaitch, W. R., Banic, C. M., Isaac, G. A., Conture, M. D., Liu, P. S. K., Gultepe, I., Li, S.-M., Kleinman, L., Daum, P. M. and MacPherson, J. I.</td>
    <td>1996</td>
    <td>Physical and chemical observations in marine stratus during the 1993 NARE: factors controlling cloud droplet number concentrations. J. Geophys. Res., 101, 29123-29135</td>
  </tr>
  <tr>
    <td>Lee, I. Y., Hanel, G. and Pruppacher, H. R.</td>
    <td>1980</td>
    <td>A numerical determination of the evolution of cloud drop spectra due to condensation on natural aerosol particles. J. Atmos. Sci., 37, 1839-1853</td>
  </tr>
  <tr>
    <td>Li, S.-M., Banic, C. M., Leaitch, W. R., Liu, P. S. K., Isaac, G. A., Zhou, X.-L. and Lee, Y.-N.</td>
    <td>1996</td>
    <td>Water-soluble fractions of aerosol and the relations to number size distributions based on aircraft measurements from the North Atlantic Regional Experiment. J. Geophys. Res., 101, 29111-29121</td>
  </tr>
  <tr>
    <td>Liou, K. N.</td>
    <td>1992</td>
    <td>Radiation and cloud processes in the atmosphere. Oxford Monographs on Geology and Geophysics, No. 20</td>
  </tr>
  <tr>
    <td>Liu, P. S. K., Leaitch, W. R., Banic, C. M., Li, S.-M., Ngo, D. and Megaw, W. J.</td>
    <td>1996</td>
    <td>Aerosol observations at Chebogue Point during the 1993 NARE: Relationships among CCN, size distribution and chemistry. J. Geophys. Res., 101, 28971-28990</td>
  </tr>
  <tr>
    <td>Meador, W. E. and Weaver, W. R.</td>
    <td>1980</td>
    <td>Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement. J. Atmos. Sci., 37, 630-643</td>
  </tr>
  <tr>
    <td>Platnick, S. and Twomey, S.</td>
    <td>1994</td>
    <td>Determining the susceptibility of cloud albedo to changes in droplet concentration with the Advanced Very High Resolution Radiometer. J. Appl. Meteorol., 33, 334-347</td>
  </tr>
  <tr>
    <td>Schwartz, S. E. and Slingo, A.</td>
    <td>1996</td>
    <td>Enhanced shortwave cloud radiative forcing due to anthropogenic aerosols. In Clouds, chemistry and climate. Eds. P. J. Crutzen and V. Ramanathan. NATO ASI Series Vol. 135. Springer Verlag, Berlin, Heidelberg</td>
  </tr>
  <tr>
    <td>Shettle, E. P. and Fenn, R. W.</td>
    <td>1976</td>
    <td>'Models of the atmospheric aerosols and their optical properties'. In AGARD conference proceedings No. 183, Optical propagation in the atmosphere. Paper presented at the Electromagnetic Wave Propagation Panel Symposium, Lungby, Denmark, 27-31 October 1975. AGARD-CP-183, AD A028-615</td>
  </tr>
  <tr>
    <td>Slingo, A.</td>
    <td>1989</td>
    <td>A GCM parameterisation for the shortwave radiative properties of water clouds. J. Atmos. Sci., 46, 1419-1427</td>
  </tr>
</table>
<table>
  <tr>
    <td>Slingo, A.</td>
    <td>1990</td>
    <td>Sensitivity of the earth's radiation budget to changes in low clouds. <i>Nature</i>, <b>343</b>, 49-51</td>
  </tr>
  <tr>
    <td>Slingo, A., Wildespin, R. C. and Smith, R. N. B.</td>
    <td>1989</td>
    <td>Effect of improved physical parameterisations on simulations of cloudiness and the earth's radiation budget in the tropics. <i>J. Geophys. Res.</i>, <b>94</b>, 2281-2302</td>
  </tr>
  <tr>
    <td>Smith, R. N. B.</td>
    <td>1990</td>
    <td>A scheme for predicting layer clouds and their water content in a general circulation model. <i>Q. J. R. Meteorol. Soc.</i>, <b>116</b>, 435-460</td>
  </tr>
  <tr>
    <td>Stephens, G. L.</td>
    <td>1978</td>
    <td>Radiation profiles in extended water clouds. II. Parameterization schemes. <i>J. Atmos. Sci.</i>, <b>35</b>, 2123-2132</td>
  </tr>
  <tr>
    <td>Zdunkowski, W. G. and Breslin, P.</td>
    <td>1979</td>
    <td>A numerical test of two approximate solutions to the radiative transfer equation using the Elsasser scheme. <i>Pure Appl. Geophys.</i>, <b>117</b>, 927-934</td>
  </tr>
  <tr>
    <td>Zdunkowski, W. G., Welch, R. M. and Korb, G.</td>
    <td>1980</td>
    <td>An investigation of the structure of typical two-stream methods for the calculation of solar fluxes and heating rates in clouds. <i>Beitr. Phys. Atmos.</i>, <b>53</b>, 147-166</td>
  </tr>
  <tr>
    <td>Zdunkowski, W. G., Panhans, W. G., Welch, R. M. and Korb, G. J.</td>
    <td>1982</td>
    <td>A radiation scheme for circulation and climate models. <i>Beitr. Phys. Atmos.</i>, <b>55</b>, 215-238</td>
  </tr>
</table>