Clouds interact with solar and terrestrial radiation and thereby influence the energy budget of the earth. The importance of ice clouds for the climate has been recognized for a long time (e.g., Liou 1986). Many questions concerning the microphysical composition of ice clouds have not been solved yet since it is difficult to measure ice crystals in nature. In general, ice clouds are found in the upper part of the troposphere, requiring aircrafts suited for this height. Low water vapor content and a low abundance of ice nuclei contribute to the low concentration of ice crystals that make their observation difficult. An exception might be tropical anvils or cirrus clouds originating thereof, whose composition and ice load is strongly controlled by the vertical transport from the lower troposphere. Another difficulty is the measurement of the ice particle size with scattering probes like the Forward Scattering Spectrometer Probe (FSSP) since nonspherical particles produce a complicated signal in the detector. Nonscattering optical probes like 2D arrays can sample irregularly shaped crystals, but their detection limit used to be high, typically on the order of some 10 μm. New techniques like the video ice particle sampler (VIPS, McFarquhar and Heymsfield 1996) or the counterflow virtual impactor (CVI, Noone et al. 1993) have been developed to sample micron-sized particles, but the available data are not conclusive yet. The size distribution is still very speculative and better data are highly desired, especially for small ice particles.
The objective of this study is to find a parameterization of re for ice clouds in large-scale models. The problem is split into two parts, namely, to define a size distribution of ice particles and to examine the effect of the different definitions for re. Existing parameterizations of the particle size distribution n(L) do not take into account small ice crystals. Therefore, a new parameterization for n(L) is suggested that includes also small particles. The suggested size distribution is a function of the macroscopic variables temperature (T) and ice water content (IWC). The different interpretations of the theoretical definition (1) lead to large differences in the resulting re. It is possible to find norming factors that allow us to compare re from the different definitions. Finally, a parameterization for re in terms of large-scale variables is sought to replace the expensive explicit calculation of re in ice clouds. A fair agreement is found between observed and calculated re.
2. Shape of ice crystals
Ice crystals occur in many different shapes or habits. Early works suggest that the ambient temperature decides the crystal type (Ono 1970), but more recent research indicates that the growth processes leading to the different habits are complicated and no simple relationship between temperature and habit exists (Dowling and Radke 1990). Despite the variety of naturally occuring habits, all crystals are treated as hexagonal columns for the remainder of this work. This simplification is chosen to make re from the present work applicable in the radiation scheme of Ebert and Curry (1992), which has been developed with the same assumption. The restriction to one habit is inconsistent with observations; nevertheless, radiation schemes suitable for large-scale models have yet to be developed for other habits than hexagonal columns.
3. Size spectrum
The difficulties of measuring the size distribution in ice clouds have been mentioned in the introduction. Several mathematical expressions for the size spectrum have been suggested based on observations. The limited number of data and the hardly understood variability of ice clouds make a universal size distribution difficult to find. However, large-scale models do not resolve the microphysics explicitly and the spectrum has to be parameterized, that is, an assumption on the shape of the distribution has to be made. A variety of suitable mathematical functions is presented below and compared to observations.
a. Γ distribution
b. Exponential distribution
c. Power-law distribution
d. Mixed distribution
Heymsfield and Platt (1984) used only particles larger than 20 μm to derive the power-law distributions. Observations show that the distribution of small particles is described fairly well with a Γ distribution (Platt et al. 1989; Moss et al. 1996; Ström et al. 1997). It is thus suggested that n(L) is a combination of a Γ distribution for small particles up to 20 μm and a power-law distribution for the larger particles.
e. Comparison of the distributions
All spectra described above are tested against observations. The dataset from Heymsfield and Platt (1984) is a composition of several observations, and two additional case studies are taken from Sassen et al. (1989). Figure 1 shows the distributions for two different temperature intervals with IWC taken from Heymsfield and Platt. All spectra are in close agreement for L larger than about 100 μm with the exception of the exponential distribution in the lower temperature range. The exponential distribution underestimates the number of large particles substantially for T below about −30°C. As stated above, the data for λexp is available only down to −25°C, which might explain the worse agreement of (11) for lower temperatures. Hence, the exponential distribution with λexp from (12) is not useful for the purpose of this work.
The mixed distribution agrees well with the power-law distribution for large L and gives slightly lower values for n(L) for small L. There are indications that the mixed distribution is more likely, especially in the small particle region. For small L the distribution must remain bounded to keep ∫ n(L) dL finite. Theory also predicts that the smaller a particle, the faster it grows by vapor deposition and, hence, the number of small particles is depleted toward larger sizes. These requirements for the shape of n(L) are fulfilled by the mixed but not by the power-law distribution.
The sum of all arguments favors the mixed distribution for the particle spectrum, and (15) is chosen to describe the distribution of ice particles for the remainder of this work. However, note that (15) represents a theoretical distribution and might differ from any actual observation, but, on the other hand, should give reasonable spectra for a variety of ambient conditions. The suggested values for the parameters B(T, IWC), ν, and λ were used to compute n(L), which was then compared to observations and showed a fair agreement. Nevertheless, the available data on particle spectra, especially for small particles, are still limited and, consequently, the settings for the parameters remain uncertain.
4. Effective radius for nonspherical particles
5. Integration limits
Any definition for re requires integrations over the size spectrum. Theoretically, the integration over L extends from 0 to ∞, but practically the limits are set to a finite upper and a nonzero lower value. The integration limits for all calculations in this work are set as 10 and 1000 μm, respectively, if not stated otherwise. The suggested values are assumed to be typical for ice clouds. The sensitivity of (20) to changes in the upper limit is low: re decreases by 20% if Lmax is set to 600 μm and increases by 20% if it is set to 1900 μm. The sensitivity to changes of Lmin is more crucial: re increases by 20% already if the lower limit is 50 instead of 10 μm. The reliability of many instruments may be doubted below typically 100 μm (e.g., Sassen et al. 1989) and many studies exclude small particles due to problems with their reliable observation. However, the small particles may make an important contribution to re as shown above, and their exclusion may lead to a serious bias in re.
Real spectra certainly have an upper limit, defined by the largest possible ice crystal. The low abundance of large particles makes their observation difficult and, consequently, the shape of the spectrum is not well defined for large particles. The upper integration limit should thus not be the size of the largest measured particle but a typical maximum size for the entire cloud.
It is difficult to measure small ice particles, and the shape of the spectrum at the lower detection limit is uncertain. New results indicate that the smallest particles in ice clouds are not a few tens of microns but rather a few microns, but the results are not conclusive yet. The setting of the lower limit is considered to be preliminary and better information about small ice crystals, from measurements or theory, is highly desired to set it properly.
6. Parameterizations for re
The different parameterizations (31), (34), and (35) are shown in Fig. 4, and the spread between them is remarkable. The parameterization suggested here yields values for re that lie between re,Ou and re,Fa. The parameterization from Ou and Liou has been developed for temperatures below −20°C, and the large re,Ou at warmer temperatures might be unrealistic. Furthermore, Ou and Liou calculate re with a spectrum where all particles smaller than 20 μm have been neglected, which gives a positive bias for re (see section 5). Including smaller particles in the derivation of re,Ou might yield values not too different from those calculated with (35), at least for T below 253 K. Unfortunately, no details of the derivation of re,Fa are known, and nothing can be said about restrictions to its use or problems with possible biases.
The differences between the parameterizations presented in this section are larger than the differences arising from different definitions for re in section 4 (cf. Figs. 2 and 4). It is highly probable that the different parameterizations have been developed with different assumptions on the size distributions. This last finding clearly shows the need for a reliable size spectrum in the calculation of re.
The parameterization (35) is tested against observed re from two recent studies, both performed in 1989 as a part of the International Cirrus Experiment (ICE). Francis et al. (1994) observed ice clouds over central Scotland (ICE 215) and over the southwestern peninsula of England (ICE 217), and Gayet et al. (1996) report on cirrus over the North Sea. The values for T and IWC in (35) are taken from the observations. As mentioned before, re cannot be measured directly but has to be derived from observed quantities. Assumptions on the shape or the distribution of ice crystals have a large influence on the value obtained for re.
Francis et al. used a 2D-C probe to measure the spectrum of particles with sizes between 25 and 800 μm. They correct the spectrum with Γ distributions for smaller and larger particles to achieve the correct IWC that was measured with another method. The corrections applied reduce re and the adjusted value for re is taken for the comparison here.
The largest Δre’s are found with the observations of Gayet et al. (1996). The large difference might arise from the FSSP-100 that was used to detect small particles. The instrument is able to measure spherical particles down to 3 μm, but its performance with nonspherical particles is quite uncertain. It is possible that the FSSP overestimates the number of small particles, or, equivalently, the size of the measured particles, thus inducing a too-low value for re. Another possible explanation for the large Δre could be an underestimation of the concentration of large particles in the observations. Gayet et al. used a 2D-C probe with a detection range between 25 and 800 μm, and no correction was applied to include possible larger particles. Expression (35) has been developed for a maximum particle size of 1000 μm and, consequently, a slightly larger value for re may result from (35).
Figure 6 displays the comparison between Δre for the suggested parameterization (35) and for the other parameterizations re,Fa and re,Ou with the same set of observations as before. The relative difference obtained with (35) is lower than that from the other parameterizations, especially if the observations from Gayet et al. (encircled in Fig. 6) are excluded from the comparison.
8. Concluding remarks
The purpose of this study is to calculate re in ice clouds based on information about the cloud’s microphysics. The crystal habit is chosen to consist of hexagonal columns, mainly to make re from this study compatible with the radiation scheme from Ebert and Curry (1992). A new size distribution n(L) is suggested, composed of a Γ distribution for small crystals and a power-law distribution for large crystals. The new n(L) is able to reproduce observed ice spectra.
With n(L) it is possible to calculate the effective radius. However, re depends on the definition used—that is, how the r3 and r2 terms in (1) are interpreted. Norming factors have been developed that allow a comparison of re from the different definitions. The norming factors are also useful for practical applications, for example, to apply any re in a given radiation scheme even if a different definition for re has been used to develop the scheme.
The assumptions on the shape, size distribution, and definition allow us to compute re explicitly, but for practical reasons a parameterization for re is sought. Unlike existing parameterizations, the suggested parameterization depends on both the amount of cloud ice and the temperature. A comparison with observations shows a fair agreement, although differences may be appreciable sometimes. It is not clear, however, if the parameterization or the observation fails since the interpretation of the observed shape and spectrum of ice crystals lead to a large uncertainty in the observed re.
The results from this study shed light on the sensitivity of re to the microphysical composition of ice clouds. There is a need for better data about the size distribution and shape of ice crystals in order to improve the retrieval for re with all its consequences for the radiation calculation. Special emphasis should be put on small particles whose reliable detection has been impossible hitherto. The small particles may make an important contribution for the radiative transfer in clouds and should not be neglected (Zender and Kiehl 1994; Arnott et al. 1994). It might be possible to gather more data from in situ observations, but the detection of small nonspherical particles is not easy.
Remote sensing techniques have a great potential to improve the knowledge about the composition of clouds (e.g., Ou et al. 1995). The transmission through and reflection in clouds can be measured directly with ground- or space-based platforms; the problem then is to solve the inverse problem, that is, to find the phase, size, and shape of the cloud particles from the observed radiation. Another promising technique is the advent of active optical sensors such as lidars—their advantage is the well-defined direction, frequency, polarization, and pulse length of the light beam, which opens new possibilities in remote sensing.
Better understanding may also come from microphysical models where crystal nucleation and growth are studied for a variety of environmental conditions. The information from these models might be used to improve the assumptions made in this study on the size distribution and, possibly, on the crystal habit.
I would like to thank Prof. H. Sundqvist and two anonymous reviewers for their critical comments that helped improve this manuscript. This study has been supported by the Swedish Natural Science Research Council (NFR) under Grant G-AA/GU02923-317.
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Ice particle number concentration (Ntot) in L−1, observed on 17 October 1983 (Sassen et al. 1989) and calculated with the distributions as given in the text. The upper and lower half of the table are for different assumptions on the lower integration limit Lmin.