## 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 *μ*m. 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.

*r*

_{e}) 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,

*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

*r*

_{e}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 *r*_{e} 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 *r*_{e}. 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 *r*_{e}. It is possible to find norming factors that allow us to compare *r*_{e} from the different definitions. Finally, a parameterization for *r*_{e} in terms of large-scale variables is sought to replace the expensive explicit calculation of *r*_{e} in ice clouds. A fair agreement is found between observed and calculated *r*_{e}.

## 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 *r*_{e} 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.

*V*) and surface area (

*A*) for a hexagonal, symmetrical column with length

*L*and width

*D*become

*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)

*L*(Pruppacher and Klett 1978):

*m*

*L*

*ρ*

*L*

*V*

*L*

^{−2}

*L*

^{4}

^{2.7625}

*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.

*n*(

*L*) =

*A*

_{x}

*n*

_{x}(

*L*), where

*n*

_{x}(

*L*) denotes a specific mathematical function describing the distribution. Since large-scale models usually provide information about the ice content the free parameter

*A*

_{x}is linked to the ice content,

### a. Γ distribution

*ν*

_{Γ}= 1, based on observed spectra from Sassen et al. (1989). Here, two more values, 0 and −1, are assigned to

*ν*

_{Γ}to extend the variability of the Γ distribution. The dataset from Heymsfield and Platt (1984) includes observed values for

*N*

_{100}=

*n*(100

*μ*m) and IWC

_{obs}as a function of temperature. To calculate IWC with (7), it is assumed that the integration over

*L*extends from 10 to 1000

*μ*m. A discussion of the choice of the integration limits is given in section 5. Values for

*λ*

_{Γ}are then found by numerically minimizing:

*λ*

_{Γ}is not very sensitive to the temperature and, thus, can be chosen independent of

*T.*The values found for

*λ*

_{Γ}are

### b. Exponential distribution

*n*(

*L*) follows an exponential distribution,

*n*

*L*

*A*

_{exp}

*λ*

_{exp}

*L*

*L*larger than about 150

*μ*m. The form of the exponential distribution is equal to the Γ distribution with

*ν*= 0, as described above, but

*λ*

_{exp}is chosen in a different way than

*λ*

_{Γ}. According to Fig. 8 in Ryan (1996),

*λ*is a function of temperature, and a visual fit to the data in the figure yields

*λ*

_{exp}

^{(278−T)/40}

^{−3}

*T*is in kelvins and

*λ*

_{exp}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

*A*

_{P}and

*B*

_{P}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

*B*

_{P}is taken directly from Heymsfield and Platt, and

*A*

_{P}used here is equal to the original

*A*multiplied by IWC.

### d. Mixed distribution

Heymsfield and Platt (1984) used only particles larger than 20 *μ*m 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 *μ*m and a power-law distribution for the larger particles.

*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

*T*is in kelvins, IWC is in g m

^{−3}, and IWC

_{0}= 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.

*α*is chosen as to make

*n*(

*L*) continuous at 20

*μ*m. The values for

*ν*and

*λ,*3 and 0.3

*μ*m

^{−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 *μ*m 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 *λ*_{exp} is available only down to −25°C, which might explain the worse agreement of (11) for lower temperatures. Hence, the exponential distribution with *λ*_{exp} from (12) is not useful for the purpose of this work.

*N*

_{tot}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

*L*

_{max}= 1000

*μ*m and a lower limit

*L*

_{min}of either 10 or 100

*μ*m. All distributions yield comparable particle concentrations in the case

*L*

_{min}= 100

*μ*m, and the results agree with the observations. Extending the integration down to 10

*μ*m 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

*μ*m 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

*μ*m (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

*r*

_{e}from equivalent surface area spheres,

*r*

_{e}is proportional to the ratio of the mass-equivalent sphere to the cross section of the particle:

*r*

_{e}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

*r*

*D*

^{2}

*L*

^{1/3}

*r*

_{e}becomes

*r*

_{e}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

*r*

_{e}are apparent. Two consequences follow from the discrepancy: 1) a direct comparison of

*r*

_{e}from different sources is not possible and 2)

*r*

_{e}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

*r*

_{e}for different radiation schemes.

*r*

_{e}might help to circumvent the problems arising from the different definitions. The idea is to have an easy means to convert the

*r*

_{e}from the different definitions into each other. Such a conversion is required to use any arbitrary

*r*

_{e}in a specific radiation parameterization. For example,

*r*

_{e,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

*r*

_{e}follows from the requirement that any definition of

*r*

_{e}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,

*r*

_{e}are

*r*

_{e,0}formally as

*r*

_{e}for spherical droplets and it follows

*r*

_{e,0}

*n*

_{EC}

*r*

_{e,EC}

*n*

_{FT}

*r*

_{e,FT}

*r*

_{e}

*r*

_{e}defined with (20). The right frame of Fig. 2 shows

*r*

_{e}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

*r*

_{e}from different definitions comparable and allow an easy adjustment of

*r*

_{e}whenever it is needed.

## 5. Integration limits

Any definition for *r*_{e} 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 *μ*m, 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: *r*_{e} decreases by 20% if *L*_{max} is set to 600 *μ*m and increases by 20% if it is set to 1900 *μ*m. The sensitivity to changes of *L*_{min} is more crucial: *r*_{e} increases by 20% already if the lower limit is 50 instead of 10 *μ*m. The reliability of many instruments may be doubted below typically 100 *μ*m (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 *r*_{e} as shown above, and their exclusion may lead to a serious bias in *r*_{e}.

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 *r*_{e}

*r*

_{e}is expensive and large-scale models use parameterizations instead. Several parameterizations for

*r*

_{e}have been developed whereof two are presented here. McFarlane et al. (1992) suggest that

*r*

_{e}should be a function of IWC:

*r*

_{e,Fa}

*X*

^{0.786}

*X*) is given by

^{−3}to yield

*r*

_{e}in microns.

*D*

_{e}in microns) depends on the temperature (

*T*

_{c}in °C):

*D*

_{e}

*T*

_{c}

*T*

^{2}

_{c}

*T*

^{3}

_{c}

*r*

_{e}and

*D*

_{e}, which is inverted here to yield

*r*

_{e}is in microns.

*r*

_{e}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

*r*

_{e}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

*r*

_{e}might be in terms of

*B.*Note however that

*r*

_{e}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

*r*

_{e}and its parameterization must be adjusted even though

*B*remains unchanged. The parameterization for

*r*

_{e}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

*r*

_{e}with (20). Finally, a relation between the initial

*B*and the obtained

*r*

_{e}is sought and approximated with a third-order polynomial,

*r*

_{e}

*B*

*B*

^{2}

*B*

^{3}

*B*(

*T,*IWC) allows us to calculate

*r*

_{e}from

*T*and IWC. A comparison between the exactly calculated and the parameterized

*r*

_{e}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 *r*_{e} that lie between *r*_{e,Ou} and *r*_{e,Fa}. The parameterization from Ou and Liou has been developed for temperatures below −20°C, and the large *r*_{e,Ou} at warmer temperatures might be unrealistic. Furthermore, Ou and Liou calculate *r*_{e} with a spectrum where all particles smaller than 20 *μ*m have been neglected, which gives a positive bias for *r*_{e} (see section 5). Including smaller particles in the derivation of *r*_{e,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 *r*_{e,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 *r*_{e} 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 *r*_{e}.

## 7. Validation

The parameterization (35) is tested against observed *r*_{e} 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, *r*_{e} 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 *r*_{e}.

Francis et al. used a 2D-C probe to measure the spectrum of particles with sizes between 25 and 800 *μ*m. 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 *r*_{e} and the adjusted value for *r*_{e} is taken for the comparison here.

*r*

_{e}, defined as

*T*or IWC. In general, Δ

*r*

_{e}is positive, which means that the parameterized

*r*

_{e}exceeds the observed. However, no general conclusion can be drawn since the variability of Δ

*r*

_{e}is high. No correlation is found for Δ

*r*

_{e}and

*T.*There might be an increase in Δ

*r*

_{e}with low IWC, but, at the same time, the variability of Δ

*r*

_{e}increases, which makes the possible trend uncertain.

The largest Δ*r*_{e}’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 *μ*m, 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 *r*_{e}. Another possible explanation for the large Δ*r*_{e} 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 *μ*m, and no correction was applied to include possible larger particles. Expression (35) has been developed for a maximum particle size of 1000 *μ*m and, consequently, a slightly larger value for *r*_{e} may result from (35).

Figure 6 displays the comparison between Δ*r*_{e} for the suggested parameterization (35) and for the other parameterizations *r*_{e,Fa} and *r*_{e,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 *r*_{e} in ice clouds based on information about the cloud’s microphysics. The crystal habit is chosen to consist of hexagonal columns, mainly to make *r*_{e} 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, *r*_{e} depends on the definition used—that is, how the *r*^{3} and *r*^{2} terms in (1) are interpreted. Norming factors have been developed that allow a comparison of *r*_{e} from the different definitions. The norming factors are also useful for practical applications, for example, to apply any *r*_{e} in a given radiation scheme even if a different definition for *r*_{e} has been used to develop the scheme.

The assumptions on the shape, size distribution, and definition allow us to compute *r*_{e} explicitly, but for practical reasons a parameterization for *r*_{e} 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 *r*_{e}.

The results from this study shed light on the sensitivity of *r*_{e} 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 *r*_{e} 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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Effective radius as a function of *T* for the different definitions of *r*_{e} 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 *r*_{e} 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

Effective radius as a function of *T* for the different definitions of *r*_{e} 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 *r*_{e} 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

Effective radius as a function of *T* for the different definitions of *r*_{e} 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 *r*_{e} 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

Exact (solid) and parameterized (dashed) *r*_{e} 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

Exact (solid) and parameterized (dashed) *r*_{e} 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

Exact (solid) and parameterized (dashed) *r*_{e} 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

Effective radius as a function of *T* for various parameterizations. The dashed line denotes *r*_{e,Ou}, the dashed–dotted *r*_{e,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 *r*_{e,Ou} is independent of IWC.

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

Effective radius as a function of *T* for various parameterizations. The dashed line denotes *r*_{e,Ou}, the dashed–dotted *r*_{e,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 *r*_{e,Ou} is independent of IWC.

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

Effective radius as a function of *T* for various parameterizations. The dashed line denotes *r*_{e,Ou}, the dashed–dotted *r*_{e,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 *r*_{e,Ou} is independent of IWC.

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

The relative error Δ*r*_{e} 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

The relative error Δ*r*_{e} 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

The relative error Δ*r*_{e} 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

As in Fig. 5, but all Δ*r*_{e} obtained with (35) are shown as stars. Additionally, results obtained with the other parameterizations *r*_{e,Fa} (×) and *r*_{e,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

As in Fig. 5, but all Δ*r*_{e} obtained with (35) are shown as stars. Additionally, results obtained with the other parameterizations *r*_{e,Fa} (×) and *r*_{e,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

As in Fig. 5, but all Δ*r*_{e} obtained with (35) are shown as stars. Additionally, results obtained with the other parameterizations *r*_{e,Fa} (×) and *r*_{e,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

Ice particle number concentration (*N*_{tot}) 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 *L*_{min}.