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  • Mitchell, D. L., 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part I: Microphysics. J. Atmos. Sci.,51, 797–816.

  • Moss, S. J., P. N. Francis, and D. G. Johnson, 1996: Calculation and parameterization of the effective radius of ice particles using aircraft data. Proc. 12th Int. Conf. on Clouds and Precipitation, Zurich, Switzerland, Int. Commission on Clouds and Precipitation and Int. Assoc. of Meteorology and Atmospheric Science, 1255–1258.

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  • ——, and Coauthors, 1995: Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: Verification using FIRE II IFO measurements. J. Atmos. Sci.,52, 4143–4158.

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  • Zender, C. S., and J. T. Kiehl, 1994: Radiative sensitivities of tropical anvils to small ice crystals. J. Geophys. Res.,99, 25 869–25 880.

  • View in gallery
    Fig. 1.

    Comparison of different calculated spectra with data for IWC and T taken from Heymsfield and Platt (1984). The left frame shows n(L) for the temperature interval −20° to −25°C, the right frame for −45° to −50°C. Shown are Γ distributions (dashed–dotted), exponential (dotted), power-law (dashed), and the mixed distribution (solid) as suggested in this work. The power-law distribution is the fit of Heymsfield and Platt to the observations and, hence, is close to the true spectra.

  • View in gallery
    Fig. 2.

    Effective radius as a function of T for the different definitions of re from Ebert and Curry (1992, dashed), from Foot (1988, dashed–dotted), and for the definition (20) suggested here (solid). The same n(L) and IWC has been used for all three curves. The left frame shows re for the three definitions, whereas it has been multiplied with the norming factors (29) and (30) in the right frame.

  • View in gallery
    Fig. 3.

    Exact (solid) and parameterized (dashed) re according to (35) for two values of IWC.

  • View in gallery
    Fig. 4.

    Effective radius as a function of T for various parameterizations. The dashed line denotes re,Ou, the dashed–dotted re,Fa, and the solid the parameterization (35) suggested in this study. Two different values, 0.1 and 10−5 g m−3, have been used for IWC. Note that re,Ou is independent of IWC.

  • View in gallery
    Fig. 5.

    The relative error Δre between (35) and observations made in three different ice clouds. The result is coded with respect to the observation: Stars for Gayet et al. (1996), crosses for ICE 215, and circles for ICE 217 (Francis et al. 1994).

  • View in gallery
    Fig. 6.

    As in Fig. 5, but all Δre obtained with (35) are shown as stars. Additionally, results obtained with the other parameterizations re,Fa (×) and re,Ou (+) are shown. No distinction has been made for the three different observations. The points encircled are from the comparison between the paramaterization (35) and the observations of Gayet et al. (1996).

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The Effective Radius in Ice Clouds

Klaus WyserDepartment of Meteorology, Stockholm University, Stockholm, Sweden

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Abstract

The effective radius (re) is a measure of the particle size used to calculate the optical properties of clouds. The objective of this study is to derive re from the microphysical composition of ice clouds. All ice crystals are assumed to be hexagonal columns with an aspect ratio depending on their size. Several existing particle size distributions are evaluated. The shape of the spectra is considered to be unsatisfactory for small particles and a new distribution is suggested that includes a Γ distribution for small crystals. The suggested spectrum agrees well with observations, although it is still speculative for small particles due to the limited availability of data.

The effective radius for nonspherical particles is not uniquely defined, and several possible definitions for re are tested. Large differences in re arise from the different definitions even if the same assumptions on the shape and the size distribution of ice particles are used. Norming factors help to adjust the differently defined re in order to make re from different sources compatible.

Finally, a new parameterization for re is suggested to avoid the expensive explicit computation. The proposed parameterization makes re a function of both the ice content and the temperature. A fair agreement between parameterized and observed re is found.

Corresponding author address: Dr. Klaus Wyser, Department of Meteorology, Stockholm University, S-106 91 Stockholm, Sweden.

Email: klaus@misu.su.se

Abstract

The effective radius (re) is a measure of the particle size used to calculate the optical properties of clouds. The objective of this study is to derive re from the microphysical composition of ice clouds. All ice crystals are assumed to be hexagonal columns with an aspect ratio depending on their size. Several existing particle size distributions are evaluated. The shape of the spectra is considered to be unsatisfactory for small particles and a new distribution is suggested that includes a Γ distribution for small crystals. The suggested spectrum agrees well with observations, although it is still speculative for small particles due to the limited availability of data.

The effective radius for nonspherical particles is not uniquely defined, and several possible definitions for re are tested. Large differences in re arise from the different definitions even if the same assumptions on the shape and the size distribution of ice particles are used. Norming factors help to adjust the differently defined re in order to make re from different sources compatible.

Finally, a new parameterization for re is suggested to avoid the expensive explicit computation. The proposed parameterization makes re a function of both the ice content and the temperature. A fair agreement between parameterized and observed re is found.

Corresponding author address: Dr. Klaus Wyser, Department of Meteorology, Stockholm University, S-106 91 Stockholm, Sweden.

Email: klaus@misu.su.se

1. Introduction

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.

If large-scale models treat ice clouds at all, they often do not take into account the varying size of the ice particles in their radiation calculation. Recently, new radiation schemes have been developed that accept the size of ice crystals as an input parameter (e.g., Ebert and Curry 1992; Fu 1996). The effective radius (re) is a measure for the mean size in a particle population. The extinction of radiation by a particle is governed by the cross-section area of the particle (Liou 1992). In the case of spherical particles (e.g., water droplets) this leads to the well-known expression for the cross-section area weighted mean radius,
i1520-0442-11-7-1793-e1
where r is the radius and n(r) is the particle distribution with respect to r. The situation becomes more complicated for nonspherical particles like ice crystals. Nonspherical particles do not have a well-defined radius; instead, their size distribution is usually defined with respect to their maximum dimension or length (L), where n(L)dL is the number of particles per unit volume with length between L and L + dL. The integration over the size spectrum becomes ∫ n(L)dL, accordingly. The average size of an ice crystal is not uniquely given by L but should also contain information about its shape. In the simplest case this information is given by the minimum dimension or width (D) of the crystal. However, the definition for re is not unambiguous even if both L and D are known since (1) is valid for spherical particles only. A multitude of possible interpretations of (1) for nonspherical particles has evolved and will be discussed in section 4.

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.

The volume (V) and surface area (A) for a hexagonal, symmetrical column with length L and width D become
i1520-0442-11-7-1793-e2
The cross-section area (C) of a randomly oriented convex particle is equal to one-quarter of its surface area and becomes for a hexagonal column (Takano and Liou 1989)
i1520-0442-11-7-1793-e4
The relationship between the length and the width of a solid column is defined as
i1520-0442-11-7-1793-e5
Expression (5) is close to the aspect ratio suggested by Ebert and Curry (1992) except that it is continuous for all crystal sizes.
The mass of an ice crystal is the product of its density with its volume; both are parameterized as functions of L (Pruppacher and Klett 1978):
mLρLVL−2L42.7625
where m is in grams and L is in microns. The parameters in (6) have been chosen for cold, solid columns with L/D > 2.

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.

All parameterizations of the size spectrum are of the form n(L) = Axnx(L), where nx(L) denotes a specific mathematical function describing the distribution. Since large-scale models usually provide information about the ice content the free parameter Ax is linked to the ice content,
i1520-0442-11-7-1793-e7
which is inverted to yield
i1520-0442-11-7-1793-e8
Note that the major contribution to the IWC comes from large crystals, whereas the optical properties (e.g., solar albedo) are strongly controlled by the small particles (Zender and Kiehl 1994). Large errors in the computed radiative fluxes are possible since any error in IWC leads to the wrong number of ice particles, either a few large or many small crystals, and the response in the optical properties is completely different for the two cases. The quality of the modeled radiation will certainly improve if information about ice nuclei and ice particle number concentration, possibly as a function of cloud age and history, becomes available.

a. Γ distribution

Mitchell (1994) describes the ice crystal spectrum with a Γ distribution,
i1520-0442-11-7-1793-e9
and suggests to set νΓ = 1, based on observed spectra from Sassen et al. (1989). Here, two more values, 0 and −1, are assigned to νΓ to extend the variability of the Γ distribution. The dataset from Heymsfield and Platt (1984) includes observed values for N100 = n (100 μm) and IWCobs as a function of temperature. To calculate IWC with (7), it is assumed that the integration over L extends from 10 to 1000 μm. A discussion of the choice of the integration limits is given in section 5. Values for λΓ are then found by numerically minimizing:
i1520-0442-11-7-1793-e10
It appears that λΓ is not very sensitive to the temperature and, thus, can be chosen independent of T. The values found for λΓ are
i1520-0442-11-7-1793-eq1

b. Exponential distribution

Ryan (1996) suggests that n(L) follows an exponential distribution,
nLAexpλexpL
in particular for particles with L larger than about 150 μm. The form of the exponential distribution is equal to the Γ distribution with ν = 0, as described above, but λexp is chosen in a different way than λΓ. According to Fig. 8 in Ryan (1996), λ is a function of temperature, and a visual fit to the data in the figure yields
λexp(278−T)/40−3
where T is in kelvins and λexp in inverse microns. The available data extends only down to −25°C, and a linear extrapolation to colder temperatures is doubtful, as will be seen later.

c. Power-law distribution

The power-law spectrum is taken from Heymsfield and Platt (1984), who fitted power functions to observed size distributions,
i1520-0442-11-7-1793-e13
where both AP and BP are functions of T. For a good agreement between observations and (13), it is necessary to split up the spectrum in two parts: one for large and one for small particles. The parameter BP is taken directly from Heymsfield and Platt, and AP used here is equal to the original A multiplied by IWC.

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.

The power-law spectrum has the same functional form as (13), but the parameter B is chosen differently. Data gathered during the Central Equatorial Pacific Experiment (CEPEX) reveal that B depends on both T and IWC. The variability of B may be described by
i1520-0442-11-7-1793-e14
where T is in kelvins, IWC is in g m−3, and IWC0 = 50 g m−3. Expression (14) has been derived from data courtesy of A. Heymsfield. The functional form of B broadens the spectrum for higher T and/or higher IWC in agreement with current understanding of cloud microphysics.
The total mixed spectrum becomes
i1520-0442-11-7-1793-e15
where α is chosen as to make n(L) continuous at 20 μm. The values for ν and λ, 3 and 0.3 μm−1, respectively, are kept constant. The limited amount of available data on the distribution of small ice crystals does not allow for a more sophisticated parameterization of the small 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.

From Fig. 1 it also follows that the concentration of small particles is lower with any of the Γ distributions compared to the power-law or mixed distribution. The difference between the distributions yields different total particle concentrations, given by
i1520-0442-11-7-1793-e16
Table 1 shows a comparison between observed (Sassen et al. 1989) and calculated Ntot from the different suggested spectra. Only one case reported by Sassen et al. is displayed in the table; the results of the comparison in the other case (8 March 1985) are comparable to the ones shown here. Equation (16) is evaluated with an upper integration limit Lmax = 1000 μm and a lower limit Lmin of either 10 or 100 μm. All distributions yield comparable particle concentrations in the case Lmin = 100 μm, and the results agree with the observations. Extending the integration down to 10 μm has only minor impact on the concentrations from the Γ distributions, but the changes are large for the power-law and mixed distributions. Sassen et al. (1989) state that their measurements were not accurate below 100 μm and probably underestimate the number of small particles. On the contrary, the power-law distribution is based on observations that are trustworthy down to 20 μm (Heymsfield and Platt 1984). These two reasons give strong evidence that the Γ distributions, which reproduce closely the observations of Sassen et al., give too-low values for the concentration of small particles, at least for the setting of ν used here.

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

The effective radius for spherical particles (1) is defined as the cross-section weighted mean radius (Liou 1992). For nonspherical particles (1) has been interpreted in various ways. Ebert and Curry (1992) calculate re from equivalent surface area spheres,
i1520-0442-11-7-1793-e17
According to Foot (1988), re is proportional to the ratio of the mass-equivalent sphere to the cross section of the particle:
i1520-0442-11-7-1793-e18
Both definitions (17) and (18) do not fulfill the requirement for re to be the cross-section weighted mean radius. Thus, a third definition is suggested here based on a proposition by Liou (1992), who assumes that the mean radius of any particle is proportional to (DL)1/2. However, this is true only for crystals with their major axis oriented perpendicular to the incident radiation. The mean radius for a randomly oriented ice particle might be better approximated by
rD2L1/3
and, hence, re becomes
i1520-0442-11-7-1793-e20
All three definitions for re are independent of the crystal habit. Their evaluation, however, requires relationships between D, A, C, V, and L that depend on the crystal shape. Using the assumptions suggested in section 2 yields the result shown in the left frame of Fig. 2. The particle size distribution used was calculated with IWC and T from Heymsfield and Platt (1984). Large differences between the different definitions for re are apparent. Two consequences follow from the discrepancy: 1) a direct comparison of re from different sources is not possible and 2) re has to be chosen properly for a specific radiation parameterization. The problem has been addressed by Francis et al. (1994), who match the optical thickness of ice clouds in order to adjust re for different radiation schemes.
Norming factors for re might help to circumvent the problems arising from the different definitions. The idea is to have an easy means to convert the re from the different definitions into each other. Such a conversion is required to use any arbitrary re in a specific radiation parameterization. For example, re,FT has to be multiplied with the proper norming factor to be used in the parameterization from Ebert and Curry (1992). A possible normalization for re follows from the requirement that any definition of re should yield the same result for D = L. Note that the particles in this special case need not be spheres; they are still hexagonal columns but with an aspect ratio of 1. With D = L, the cross section, surface area, and volume defined in section 2 become, respectively,
i1520-0442-11-7-1793-e21
and, subsequently, the different re are
i1520-0442-11-7-1793-e24
Define re,0 formally as
i1520-0442-11-7-1793-e27
based on the definition of re for spherical droplets and it follows
re,0nECre,ECnFTre,FTre
where the norming factors are
i1520-0442-11-7-1793-e29
Note that no norming factor is necessary for re defined with (20). The right frame of Fig. 2 shows re as a function of T for the different definitions, but now multiplied with the norming factors. The difference between the three curves is greatly reduced compared to the left frame of Fig. 2 but does not vanish entirely since the equality stated in (28) is valid only for the case D = L. Nevertheless, the norming factors help to make re from different definitions comparable and allow an easy adjustment of re whenever it is needed.

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 explicit calculation of re is expensive and large-scale models use parameterizations instead. Several parameterizations for re have been developed whereof two are presented here. McFarlane et al. (1992) suggest that re should be a function of IWC:
re,FaX0.786
where the mean crystal length (X) is given by
i1520-0442-11-7-1793-e32
with IWC in g m−3 to yield re in microns.
According to Ou and Liou (1995), the mean effective size (De in microns) depends on the temperature (Tc in °C):
DeTc0.197T2c0.0012T3c
Ou et al. (1995) suggest a relation between re and De, which is inverted here to yield
i1520-0442-11-7-1793-e34
where re is in microns.
The fundamental difference between the two parameterizations is that re is related only to IWC in (31) and only to T in (34). In the present work n(L) is assumed to depend on both IWC and T and, consequently, the parameterization of re should take this double dependency into account. The only step in the derivation of n(L) where T and IWC appear together is (14), and, thus, the parameterization of re might be in terms of B. Note however that re also depends on the distribution of small particles, which is assumed to be independent of IWC and T. Any change in the small particle spectrum induces a change in re and its parameterization must be adjusted even though B remains unchanged. The parameterization for re is found by first calculating n(L) according to (15) as a function of B in the range between −6 and −2; this n(L) is then used to calculate re with (20). Finally, a relation between the initial B and the obtained re is sought and approximated with a third-order polynomial,
reBB2B3
Expression (35) together with (14) for B(T, IWC) allows us to calculate re from T and IWC. A comparison between the exactly calculated and the parameterized re is shown in Fig. 3.

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.

7. Validation

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 relative difference between parameterized and observed re, defined as
i1520-0442-11-7-1793-e36
is shown in Fig. 5 as a function of T or IWC. In general, Δre is positive, which means that the parameterized re exceeds the observed. However, no general conclusion can be drawn since the variability of Δre is high. No correlation is found for Δre and T. There might be an increase in Δre with low IWC, but, at the same time, the variability of Δre increases, which makes the possible trend uncertain.

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.

Acknowledgments

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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Fig. 1.
Fig. 1.

Comparison of different calculated spectra with data for IWC and T taken from Heymsfield and Platt (1984). The left frame shows n(L) for the temperature interval −20° to −25°C, the right frame for −45° to −50°C. Shown are Γ distributions (dashed–dotted), exponential (dotted), power-law (dashed), and the mixed distribution (solid) as suggested in this work. The power-law distribution is the fit of Heymsfield and Platt to the observations and, hence, is close to the true spectra.

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Fig. 2.
Fig. 2.

Effective radius as a function of T for the different definitions of re from Ebert and Curry (1992, dashed), from Foot (1988, dashed–dotted), and for the definition (20) suggested here (solid). The same n(L) and IWC has been used for all three curves. The left frame shows re for the three definitions, whereas it has been multiplied with the norming factors (29) and (30) in the right frame.

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Fig. 3.
Fig. 3.

Exact (solid) and parameterized (dashed) re according to (35) for two values of IWC.

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Fig. 4.
Fig. 4.

Effective radius as a function of T for various parameterizations. The dashed line denotes re,Ou, the dashed–dotted re,Fa, and the solid the parameterization (35) suggested in this study. Two different values, 0.1 and 10−5 g m−3, have been used for IWC. Note that re,Ou is independent of IWC.

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Fig. 5.
Fig. 5.

The relative error Δre between (35) and observations made in three different ice clouds. The result is coded with respect to the observation: Stars for Gayet et al. (1996), crosses for ICE 215, and circles for ICE 217 (Francis et al. 1994).

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Fig. 6.
Fig. 6.

As in Fig. 5, but all Δre obtained with (35) are shown as stars. Additionally, results obtained with the other parameterizations re,Fa (×) and re,Ou (+) are shown. No distinction has been made for the three different observations. The points encircled are from the comparison between the paramaterization (35) and the observations of Gayet et al. (1996).

Citation: Journal of Climate 11, 7; 10.1175/1520-0442(1998)011<1793:TERIIC>2.0.CO;2

Table 1.

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.

Table 1.
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