Infrared properties of cirrus clouds in climate ... - Wiley Online Library

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: 273–282 (2007) Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.1

Infrared properties of cirrus clouds in climate models Ruben Rodriguez De Leon* and Joanna D. Haigh Space and Atmospheric Physics, Blackett Laboratory, Imperial College, London, UK

ABSTRACT: The representation in global climate models of the infrared radiative properties of cirrus clouds is assessed by comparing their radiative forcing calculated using airborne in-situ-measured size distributions and retrievals from combined lidar and Doppler-radar data. The latter are fitted to a bimodal function, allowing the inclusion of the size distribution’s shape, normally omitted in the characterization of cirrus. The impact of the particle size distribution’s shape on the atmosphere’s radiation fields is evaluated using a two-stream radiative code. The comparisons show that the effect of the shape of the size distributions used to calculate the radiative forcing of a cirrus layer composed of hexagonal cylinders is not negligible, evidencing the ambiguity linked to the commonly used two-parameter (effective radius and ice water content) characterization of cirrus, and showing that the inclusion of a simple measure of the relative concentration of small particles improves its radiative parameterization. Copyright  2007 Royal Meteorological Society KEY WORDS

bimodal size distribution; ice crystals; scattering; optical properties; radiative transfer; T-matrix

Received 17 August 2005; Revised 2 March 2006; Accepted 5 October 2006

1.

Introduction

Cirrus clouds represent one of the most complex components of the climate system. The uncertainties in the characterization and modelling of their radiative properties are mainly linked to the complex shapes of their particles and the difficulties involved in the detection of very small crystals. The large range of size parameters observed in cirrus and the lack of a coordinate system in which Maxwell’s equations can be naturally solved has limited the use of realistic shapes in their representation. Most global climate models (GCMs) incorporate a plane-parallel approach in their radiative-transfer calculations, and represent liquid-water clouds with only two variables, the effective radius (re ) and the liquid-water content (LWC). Liquid-water clouds do not generally present multimodal particle size distributions (PSDs) nor large dispersion around their modal size, making it possible to use one equivalent size to characterize the ensemble. This representation has been inherited by ice clouds despite their frequently presenting bimodal distributions and large dispersions, which make the dependence of their radiative properties more sensitive to the shape of the PSD and produce ambiguous relations between the cirrus crystal sizes and its bulk single-scattering properties. Mitchell (2002) found uncertainties reaching 30% for the absorption efficiency and 48% for the extinction efficiency in the window region due to the differences in * Correspondence to: Ruben Rodriguez De Leon, Centre for Air Transport and the Environment, Manchester Metropolitan University, Faculty of Science and Engineering, Chester Street, Manchester, M1 5GD, UK. E-mail: [email protected] Copyright  2007 Royal Meteorological Society

the shape of cirrus PSDs. Previously, Zhang et al. (1999) compared the effect of using single-modal and bimodal size distributions based on field measurements by examining the percentage of ice particles smaller than 100 µm in the total particle concentration, and found that bimodal size distributions (with larger percentages of small particles) tend to have a smaller warming effect, or even a cooling effect when the size at the second maximum is relatively large. Han et al. (2005) calculated the impact of the width of several measured size distributions on the radiative properties of ice clouds, and concluded that theoretical distributions with an explicitly assumed variance should be used in satellite retrievals. In the present study, we incorporate a dataset based on bimodal PSDs according to Donovan (2003), which provides information about the PSD’s shape and a natural way to estimate the contribution by small particles. Donovan’s bimodal gamma distributions are retrieved from lidar–radar reflectivities and Doppler mean-velocity measurements, giving a theoretical characterization of the small-particle mode for which field measurements involve large uncertainties. We compare the infrared radiative forcing (RF) calculated with the bimodal distributions and with 30 in-situ-measured PSDs, which were extrapolated by Fu (1996) to include the contribution by small particles due to the limitations to measure them in situ. Fu (1996) applied these PSDs to integrate the infrared radiative properties of hexagonal cylinders calculated with a composite method (CM) which interpolates the results obtained with the finite-difference time domain (FDTD) technique and the geometric-optics method (GOM). For comparison purposes, in our study we also assume hexagonal cylinders as a representative geometry

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for cirrus crystals but we use Baran et al.’s (2002) database which applies Havemann and Baran’s (2001) extension of the T-Matrix method (T-M) to treat nonaxisymmetrical habits, including hexagonal cylinders. In section 2, the available approaches to calculate the infrared optical properties of non-spherical habits are described, and the hexagonal cylinder radiative properties calculated with T-M are presented in section 3. The integration of these calculations and their parametrization in terms of re and ice water content (IWC), according to Fu (1996), are described in section 4. The PSD dataset based on remote retrievals fitted to a bimodal PSD model (presented in section 5) is used to include the relative concentration of small crystals and their radiative contribution computed with a radiative-transfer code described in section 6. The effect of the PSD’s shape on cirrus RF is assessed in section 7 by comparison of the integrations of the T-M results over in-situ measurements and remote retrievals of cirrus PSDs. In section 8 we draw conclusions with regard to the PSD’s shape influence on the radiative modelling of cirrus. 2.

Light scattering by non-axisymmetric particles

Scattering and its angular dependence are functions of the particle’s shape, refractive index and size parameter (x = 2πr/λ), where r is the radius of a sphere with an equivalent area or volume, and λ is the incident wavelength. Cirrus clouds are composed almost exclusively of non-axisymmetrical ice crystals. These geometries have the disadvantage of not having a natural coordinate system in which the radiative calculations can be simplified. Great effort has been put into developing efficient ways to calculate the infrared optical properties of realistic shapes with exact methods. Because of the wide range of particle sizes observed in cirrus, there is no exact or approximate method to calculate their single-scattering properties (ssp) able to cover the whole atmospheric infrared spectrum. For this reason, many approximate geometries have been used to represent the non-spherical nature of cirrus crystals, including equivalent spheres and axisymmetrical habits like cylinders and Chebyschev particles. Various approximate radiative methods have been used to treat nonspherical shapes like the anomalous diffraction theory (ADT) (van de Hulst, 1957), which is a simplification of Mie theory, and the ray-tracing GOM, which is an approximation valid in the asymptotic limit of large size parameters. The merit of ADT is its simplicity and efficiency in numerical computation. Relevant processes not included in ADT can be incorporated in a modified version that yields absorption efficiencies with errors smaller than 10% (Mitchell, 2000); however, the asymmetry factor or phase function information cannot be obtained from ADT. The GOM (Takano and Liou, 1989), on the other hand, is estimated to be valid only for size parameters larger than 30, being inappropriate for small ice crystals in the infrared. Copyright  2007 Royal Meteorological Society

Yang and Liou (1996) developed the concept of a unified approach (UA) method to treat non-spherical geometries, combining the exact FDTD technique (Yee, 1996) for small particles and an improved geometric optics method (IGOM), which covers size parameters as small as 15. Even when the FDTD technique has a practical computational limit for size parameters larger than 20, the IGOM inaccuracies produce a discontinuity when switching from one method to the other due to the large-ice absorption in the infrared, which enhances the tunnelling or above-edge effect (Bohren, 1994), not accounted for by the IGOM. The tunnelling effect is defined as the dilation of the absorption cross-section beyond the particle’s geometrical cross-sectional area produced by the deflection of radiation passing close to the particle’s edge. Approximate methods used to calculate the tunnelling effect of non-spherical habits are often based either on the complex angular momentum (CAM) theory developed by Nussenzveig and Wiscombe (1987) or on a combination of Mie theory, which overestimates the tunnelling effect of non-spherical habits (Fu et al., 1998; Liu et al., 1998) and the GOM, which neglects it. The composite method (CM), developed by Fu et al. (1998) for hexagonal cylinders, follows the ideas of the UA, using a linear combination of Mie equivalents and GOM to interpolate the FDTD calculations for small size parameters and the GOM results at the asymptotic limit. All the exact techniques for computing scattering by non-spherical particles are based on solving Maxwell’s equations either analytically or numerically. The exact methods to treat arbitrarily shaped particles with small size parameters include FDTD, the finite-element method (FEM) and the T-Matrix method (T-M). FEM and FDTD are limited to small size parameters and do not reach high nor controllable numerical accuracy. The T-M is highly accurate, fast, and applicable to particles with larger size parameters. The T-M method was initially introduced by Waterman (1971) as a technique for computing electromagnetic scattering by single homogeneous nonspherical particles, based on the Huygens principle, by expanding the incident and the scattered waves in vector spherical wave functions and relating these expansions using a T-Matrix. Exact scattering solutions for non-spherical particles using T-M have been available only for infinite cylinders (Rayleigh, 1918; Wait, 1955), spheroids (Asano and Yamamoto, 1975) and rotationally symmetric nonspherical particles (Mishchenko and Travis, 1994). In general, T-M can be applied to any particle shape, although T-M computations are much simpler and more efficient for bodies with rotational symmetry. Accordingly, almost all existing T-M computing codes assume shapes like spheroids and Chebyshev particles, with no sharp edges except for the case of finite circular cylinders. The loss of efficiency for particles with large aspect ratios is the main drawback of T-M, but new procedures (Havemann and Baran, 2001) have improved the numerical stability of T-M computations to include Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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shapes lacking axial symmetry, giving an exact solution including the 20 to 30 size-parameter region in which only approximate techniques have been used previously. In section 3 the T-M database (Baran et al., 2002) used in our study is described.

3.

The T-M/CAM database

Despite the computational advantages of T-M over other exact techniques, the calculations for large sizes and asphericity become computationally onerous. For this reason Baran et al. (2002) included results based on CAM to complement the T-M calculations in the asymptotic limit, in which the results become independent of the particle’s shape (Baran and Havemann, 1999, 2000a), leaving the tunnelling region largely covered by the exact T-M calculations, and showing a smooth transition across the two solutions (Baran et al., 2000b). Baran et al.’s database includes the infrared (3.3 to 30 µm) ssp (the extinction cross-section, the singlescattering albedo, and the asymmetry parameter) of randomly oriented hexagonal cylinders with a resolution of 0.2 µm (from 3.3 µm to 17 µm) and 0.3 µm (from 17 µm to 30 µm). This spectral resolution is enough for the calculation of the broadband radiative effect of cirrus on the fluxes and heating rates needed in GCMs. The database corresponds to Yang et al. (2000), covering column lengths from 4 to 3500 µm, discretized at the centres of 24 size bins as shown in Table I. The width of the ice columns, D, is related to their length, L, based on the relationships given by Mitchell and Arnott (1994) and Auer and Veal (1970) as

D/2 =

Table I. Database discretization. L

Lmin

Lmax

D

L

D/L

4.0 7.5 15.0 25.0 35.0 45.0 60.0 80.0 100.0 130.0 175.0 225.0 275.0 350.0 450.0 550.0 650.0 750.0 900.0 1150.0 1400.0 1750.0 2500.0 3500.0

3 5 10 20 30 40 50 70 90 110 150 200 250 300 400 500 600 700 800 1000 1300 1500 2000 3000

5 10 20 30 40 50 70 90 110 150 200 250 300 400 500 600 700 800 1000 1300 1500 2000 3000 4000

2.80 5.25 10.50 17.50 24.50 31.50 42.00 56.00 68.32 82.23 95.21 106.38 115.60 127.73 141.75 154.02 165.05 175.13 188.86 209.03 226.76 248.71 288.28 331.38

2 5 10 10 10 10 20 20 20 40 50 50 50 50 100 100 100 100 200 300 200 500 1000 1000

0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.68 0.63 0.54 0.47 0.42 0.36 0.31 0.28 0.25 0.23 0.21 0.18 0.16 0.14 0.11 0.09

Discretization of ice crystal size distributions (µm) used by Baran et al. (2002) in terms of the cylinder length, L, and width, D.

refractive index (mr ) determines the phase speed inside the medium, whereas the imaginary part (mi ) is related

0.35L −4.2395 + 0.501L − 0.00117L2 5.65L0.414

Particles smaller than 100 µm are represented with a constant aspect ratio (D/L), equal to 0.7, based on the fact that their optical properties in the infrared do not vary substantially for aspect ratios between 0.7 and 1.0, and on the assumption that small crystals cannot deviate much from a spherical shape. The aspect ratio for lengths greater than 100 µm follows the definition given by Auer and Veal (1970), and represents a monotonically decreasing function. The aspect ratios used by Baran et al. (2002) are essentially equivalent to the five discrete steps used by Fu et al. (1998) in his infrared parametrization of cirrus. The column lengths used by Fu et al. are also comparable to the ones used in our calculations, covering a range from 2 to 3100 µm, but discretized in 30 bins instead of 34. The values of the refractive indices used in Baran’s database and in Fu’s calculations correspond to Warren’s (1984) compilation. The refractive index is defined as a complex function of the wavelength (λ) given as m(λ) = mr (λ) − mi (λ)i, shown in Figure 1. The real part of the Copyright  2007 Royal Meteorological Society

L < 100 µm 100 µm ≤ L ≤ 200 µm . L ≥ 200 µm

(1)

to the bulk absorption coefficient by kabs = 4πmi /λ. The most evident feature of mi is the absorption band around 12 µm, related to a rotational oscillation mode (libration) of the ice molecule. Scattering, on the other hand, is related to mr , presenting a minimum around 11 µm, where mr gets closer to the air’s value, letting the radiation pass without deflection (Christiansen’s effect). The hexagonal cylinder ssp taken from Baran’s database are plotted in Figure 2 as a function of size and wavelength. The absorption efficiency (σa ), defined as the ratio of the absorption cross-section to the projected area (both, considering random orientation averages) shows the signature of the 6 µm and 12 µm absorption bands at small size parameters before it reaches values slightly smaller than unity (asymptotic limit) due to external reflection. Significant amounts of tunnelling (σa > 1) are present at size parameters between 1 and 10 at wavelengths with large absorption. The scattering efficiency (σs ) shows the signature of Christiansen’s band (Yang et al., 1997) around 11 µm, Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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R. R. DE LEON, J. D. HAIGH (a) Real part of the refractive index

(b) Imaginary part of the refractive index 0.5

1.6

0.4 1.4 mr

mi

0.3 0.2

1.2 0.1

1.0

0.0 5

10

15

20

25

30

5

10

20

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30

l (mm)

l (mm)

Figure 1. Refractive index of ice in the infrared (Warren, 1984).

25

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5

1

10 100 1000 Maximum Length “L” (microns)

Wavelength (microns)

20

15

10

1


15

10

1

10000

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20

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10 5

10000

10 100 1000 Maximum Length “L” (microns) Asymmetry parameter

30

25

5

20

5

10000

Single scattering albedo

30


Scattering efficiency 30



Absorption efficiency 30

1


10000

Figure 2. Single-scattering properties of hexagonal cylinders from Baran et al. (2002).

whereas the single-scattering albedo (ω ≡ σscat /σext ) evidences that only very small particles have predominant absorption. Scattering becomes clearly dominant at size parameters slightly larger than unity outside Christiansen’s band. In the far infrared, resonance produces a scattering oscillation due to the interference between diffracted and transmitted rays. Finally, the asymmetry parameter shows that the averaged scattering direction is always positive (forward scattering), with the backscattering contribution being significant only for very Copyright  2007 Royal Meteorological Society

small particles. In our study we integrated these T-M ssp and compared them against Fu et al.’s parametrization based on the CM. The ssp integrations and their representation in terms of an effective size and the ice water content is described in section 4. 4.

Size integrations

Radiative-transfer calculations are often performed assuming that the cloud is a horizontally and vertically Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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uniform plane-parallel layer. GCMs often make use of a plane-parallel approach like the two-stream approximation in which cirrus cloud layers and their radiative properties are normally specified by the IWC and re (e.g. Curry and Ebert, 1992; Wyser and Yang, 1998; Fu, 1996; Fu et al., 1998; Yang et al., 2001). The first and last two of these studies treated terrestrial radiation, and all assumed that the ice cloud’s radiative properties can be described in terms of the IWC and an effective size. In this study, we use the radiative-transfer two-stream code developed by Edwards and Slingo (1996) to calculate the RF caused by a cirrus layer. A subroutine of the Edwards and Slingo radiation code (E&S) uses re polynomials of the integrated radiative properties to parametrize the optical properties of cirrus. We modified this subroutine to include Fu et al.’s (1998) approach using their generalized size (Dge ) instead of re . The idea of an effective size is based on the concept of the photon path (Bryant and Latimer, 1969), defined as the distance that a photon travels through the particle without internal reflection or refraction occurring. This effective distance (de ) depends on the orientation of the particle but, in the case of a random orientation average, de can be approximated by the ratio of the volume and the projected area as de =

M , ρi Pa

(2)

where M is the mass, ρi is the density of ice, and Pa is the particle’s projected area. Following the ideas of the effective distance, Fu (1996) defined for hexagonal cylinders a generalized size Lmax DDLN (L) dL Lmin Dge = , (3) √ 2 Lmax 3D DL + N (L) dL 4 Lmin in which N (L) denotes the number density per unit volume, Lmin and Lmax define the range of the maximum dimension of the particles present in the distribution, and D is the length of the diagonal of the hexagonal face. In the definition of Dge the constants have been dropped, meaning that its value is only proportional to the effective distance. The generalized size indirectly accounts for the aspect ratio (D/L) of the particles in the distribution by reducing the dependence of the absorption and scattering coefficients on the habit. In the definition of Dge the contribution of small particles is relatively small, thus the largest weight in defining Dge is for moderate or large ice crystals. For realistic size distributions the value of Dge is normally larger than the mean maximum dimension due to the dominant presence of small particles. The IWC for the hexagonal cylinder geometry (Fu, 1996) is defined as √ Lmax 3 3 DDLN (L) dL. (4) ρi IWC = 8 Lmin Copyright  2007 Royal Meteorological Society

Fu et al.’s (1998) infrared parametrization for the extinction coefficient, βext , the absorption coefficient, βabs , and the asymmetry parameter, g, is given by βext a1 a2 = a0 + + 2 , IWC Dge Dge βabs Dge 2 3 = b0 + b1 Dge + b2 Dge + b3 Dge , IWC 2 3 + c3 Dge . g = c0 + c1 Dge + c2 Dge

(5) (6) (7)

The units for βext and βabs are m−1 , and an ice density of 0.9167 g cm−3 is used. The coefficients ai , bi , and ci are obtained using a minimum squares fit for each band, weighted with the Planck function at the cloud’s temperature. Fu et al.’s (1998) parametrization follows the ideas of Slingo (1989), Hu and Stamnes (1992) and Fu and Liou (1993) using 12 bands to represent the infrared region of the spectrum. The largest differences between FDTD and the CM fit are 2%, 3%, and 15% for the asymmetry parameter, the absorption coefficient and the extinction coefficient, respectively (Fu et al., 1998). We used this approach to integrate Baran et al.’s (2002) T-M/CAM calculations over the same field PSD measurements compiled by Fu (1996) and over a set of remotely retrieved PSDs described in the following section.

5.

Size distributions

In order to develop a parametrization applicable to a wide range of cases, Fu et al. (1998) integrated the CM results over 28 ice-crystal size distributions based on in-situ aircraft observations from mid-latitude and tropical regions, which were extrapolated down to 1 µm to take into account the effect of small ice crystals not detected by the instruments. Mitchell (2002) showed that the errors in the radiative parametrization of βext and βabs linked to a ‘two-parameter’ representation, like the one used by Fu et al., were unacceptable due to the relatively high concentrations of small ice crystals in cirrus clouds, making it necessary to include some information about the PSD’s shape. In order to assess the effect of the cirrus size distribution on their radiative forcing we integrated the T-M optical properties over a set of bimodal PSDs based on the effective radii retrievals reported by Donovan (2003), and over the 28 PSDs used by Fu (1996) to produce two parametrizations that we included in the E&S. Donovan’s PSD database is based on lidar and Doppler-radar data from the Atmospheric Radiation Measurement Program’s Southern Great Plains (ARM-SGP) site in central USA. The combined lidar–radar technique uses the dependence of the radar reflectivity on the squared mass of the distribution, and that of the lidar’s optical extinction on the total cross-sectional area of the cloud. Taken separately, these relationships are independent of the details of the size distribution and the Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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particles’ habit, but the combined lidar–radar relationship between the effective radius (re ), the optical extinction, and the radar reflectivity does depend on the ice-crystal habit and the exact form of the ice PSD. Donovan proposed a bimodal gamma distribution model to fit the lidar–radar measurements using four representative shapes: rosettes, planar polycrystals, compact polycrystals and complex polycrystals, but only the last two matched the model given as, Ns 1 dN (r) = dr rm,s (γs )

r

(γs −1)

− r e rm,s

rm,s r Nl 1 r (γl −1) − rm,l + e , rm,l (γl ) rm,l

(8)

where r = L/2, L is the maximum dimension of the particle, γs , γl , rm,s and rm,l are the width parameters and the modal radii for the small and large modes of the distribution, respectively. Ns and Nl correspond to the number concentrations of the small and large modal values, which can be used to quantify the relative concentration of small and large particles. Donovan’s model uses fixed values for the smallparticle mode parameters of the size distribution (Rm,s = 20.1623, γs = 4.24), where Rm,s = rm,s (γs + 2). The large-particle modal width is also fixed (γl = 3.64), and the remaining parameters, rm,l , Nl , and Ns , are parametrized in terms of the temperature and the IWC. The cross-sectional absorptions calculated with Donovan’s number concentration ratios (Nl /Ns ), and Baran et al.’s (2002) T-M optical properties for a wavelength of 12 µm are presented in Figure 3 for an IWC = 10 mg m−3 , depicting the relative absorption contribution by particles smaller than 100 µm at different temperatures. The detection of these particles made in situ involves large uncertainties.

Seven values for the IWC and for temperature, T , (see Table II) were used to produce 49 bimodal gamma size distributions for complex polycrystals, taking Donovan’s IWC and relative concentration densities for the small and large modes and modifying Nl until the IWC predicted by Donovan coincided with the IWC calculated with the hexagonal cylinder geometry. Donovan’s database is valid for not only one habit and, given the fact that our aim is to evaluate the dependence of the radiative properties of cirrus on the PSD’s shape, our approach assumes that Donovan’s PSD retrievals for polycrystals could be applied to hexagonal cylinders by equating the IWC and maximum dimensions. Unfortunately a database of the infrared properties of complex polycrystals is not available to us at the present time. The generalized size of these distributions was then calculated using Fu’s (1996) formula for hexagonal cylinders (Equation (3)). The values calculated for Dge cover a range from 22.6 µm to 156.6 µm (Table II), which corresponds well with Fu’s values. Donovan’s retrievals showed that the re values calculated with complex and compact polycrystals have differences of the order of 30%, even when either habit can explain the dataset’s average behaviour. This means that, despite the fact that certain habits could be ruled out, there is still a range of habits that would be consistent with the lidar–radar data, and the same may be said about the assumed values of the fixed parameters of the smallparticle mode. The effect of using Donovan’s PSDs on the estimated radiation fields is described in section 7.

6.

Radiative-transfer calculations

In general, the FDTD calculations used by Fu et al. (1998) to fit his CM linear combination of approximate techniques are in good agreement with

Area (sq. micron/micron/m cubed)

108 107 106 105 104 103 102

1

10

100

1000

10000

maximum dimension (microns) Figure 3. Absorption cross-sectional area of the size distributions calculated with the T-M results and Donovan’s PSD parametrization for an IWC of 10 mg m−3 and a range of temperatures from 0 ° C (dark solid line) to −60 ° C (dark dashed line). Copyright  2007 Royal Meteorological Society

Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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Table II. Lidar-radar particle size-distribution parameters. IWC (g m−3 )

0.001

0.0033

0.010

0.033

0.10

0.33

1.0

Temperature (° C)

Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns Dge Rm,l Nl /Ns

−60

−50

−40

−30

−20

−10

0

24.4 59.68 0.007934 24.0 56.81 0.007603 23.6 54.16 0.007328 23.4 51.30 0.007068 23.0 48.64 0.006859 22.8 45.77 0.006669 22.6 43.12 0.006523

28.4 78.76 0.01089 30.4 85.85 0.01222 32.3 92.42 0.01350 34.7 99.51 0.01493 36.9 106.1 0.01627 39.6 113.2 0.01773 42.1 119.7 0.01909

34.2 97.85 0.01459 40.2 114.9 0.01809 46.3 130.7 0.02134 53.2 147.8 0.02478 59.8 163.5 0.02789 66.8 180.6 0.03114 73.1 196.4 0.03408

40.9 117.0 0.01851 51.7 143.9 0.02402 62.1 169.0 0.02894 72.8 195.9 0.03400 82.4 221.0 0.03851 92.0 248.0 0.04321 100.4 273.0 0.04747

48.5 136.0 0.02243 63.7 172.9 0.02970 77.2 207.2 0.03605 90.7 244.1 0.04256 102.2 278.4 0.04838 113.3 315.3 0.05449 122.7 349.6 0.06005

61.1 155.1 0.02625 82.3 202.0 0.03510 100.2 245.5 0.04279 116.8 292.4 0.05070 130.3 335.9 0.05783 143.1 382.7 0.06536 153.7 426.3 0.07226

64.1 174.2 0.02994 86.1 231.0 0.04027 103.8 283.7 0.04927 120.3 340.6 0.05859 133.6 393.3 0.06704 146.2 450.1 0.07600 156.6 502.9 0.08424

Generalized sizes obtained with Donovan’s modal values and relative concentration densities for seven temperatures and ice water contents (IWC) (see text).

Baran’s T-M/CAM results, even in the region with largest tunnelling contribution. Havemann and Baran (2001) reported that T-M produced larger absorption and extinction efficiencies at 12.99 µm compared to FDTD, with the differences being smaller than 3% and 1%, respectively, at size parameters greater than two. These differences between the two methods are smaller than the 3% and 15% differences between the CM and the parametrization’s fit reported by Fu et al. (1998). The radiative-transfer calculations were performed in our study using the Edwards and Slingo two-stream code, covering wavelengths from 3.3 to 100 µm divided in nine bands. The effects of water vapour, ozone and CO2 and temperature were included using profiles corresponding to midlatitude summer average values. The cloud was located at an altitude between 10.5 km and 8.5 km and a vertically homogeneous 10 mg m−3 IWC profile following the case defined by Zhang et al. (1999). The cloud RF was calculated as the difference between the cloud-disturbed radiative balance and that of a cloudless atmosphere and defined as the change in the long-wave radiative flux (downwelling minus upwelling) at the top of the troposphere due to the cloud’s presence.

7.

Effect of the size distributions

The large dependence of the ssp on the particle’s size (Figure 2) suggests the importance of the size distribution’s properties on the radiative parametrization of cirrus. In order to assess the effect of the Copyright  2007 Royal Meteorological Society

PSD’s shape on the parametrized radiative properties of cirrus, the TM/CAM ssp integrated over the lidar–radar size distribution set, and over Fu’s compilation, were incorporated in the Edwards and Slingo radiative code. The long-wave radiative forcings due to the presence of the cirrus layer described in the previous section are presented in Figure 4 for Donovan’s and Fu’s PSD datasets, showing the largest differences at large generalized sizes where the lidar–radar results show RF values 10 W m−2 smaller. The opposite occurs at small generalized sizes but the differences being not as large. Given that these RFs correspond to PSDs with the same IWC and generalized sizes, these differences evidence the ambiguity produced by the two-parameter representation and the impact of the size distribution’s shape on the radiative representation of cirrus. The relative number concentration ratio (Nl /Ns ) of the bimodal PSDs used in our study, provides information about the PSD’s shape and a way to produce an unambiguous relation between the microphysical and the radiative properties of cirrus, unfortunately the comparison of the contribution by small particles with other PSD datasets not based on a bimodal model is not straightforward. For this purpose Zhang et al. (1999) related the percentage of particles smaller than 100 µm (Psm) to the radiative properties of single-mode and a bimode PSDs for spherical particles, and showed that the largest differences were found in the infrared RF, with values as large as 20 W m−2 . Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj

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Cloud Radiative Forcing (W per sq metre)

160

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Generalized size (microns)

Figure 4. Cirrus infrared radiative forcing depending on the generalized size for Fu’s (triangles) and Donovan’s PSDs (crosses). 1.05

1.00

Psm

0.95

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0

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140

Generalized size (microns)

Figure 5. Percentage of ice crystals smaller than 100 microns (Psm) in Fu’s (triangles) and Donovan’s (crosses) PSDs.

Zhang et al.’s single-mode PSDs presented Psm values smaller than 10% and for the same mean size the bimodal Psm reached values larger than 80%, this range is not necessarily representative of the measured PSDs they characterize but follows from fitting PSDs with the same IWC and mean crystal size as the field measurements using single-mode and bimodal functions. Zhang et al. found that the infrared RF was inversely proportional to the Psm. Our calculations confirm this behaviour for cirrus composed of hexagonal cylinders, as deduced from Figures 4 and 5. Given that the steep gradient shown by the absorption and scattering cross sections of hexagonal cylinders at sizes smaller than 100 µm is not exclusive of this habit, it is expected that other shapes used to represent cirrus crystals will Copyright  2007 Royal Meteorological Society

also be influenced by the relative number of small particles. The range covered by Donovan’s reported numberconcentration ratios extends from 0.006 and 0.084 for complex polycrystals and from 0.002 to 0.04 for compact polycrystals, with corresponding Psm values larger than 99% (Figure 5), which contrasts with the large Psm range reported by Zhang et al. for their bimodal model. Figure 5 shows a clear agreement between the Psm of most in-situ-measured PSDs with the lidar–radar retrieved Psm values (∼99%) with more apparent differences at generalized sizes larger than 55 µm, where in-situ-measured PSDs with less relative small-particle concentrations appear. This behaviour does not seem to be linked to the latitude at which the PSDs were Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj


measured, but more extensive remote retrievals at midlatitude and tropical sites would be needed in order to determine a representative relative concentration of small particles. The RF differences at generalized sizes smaller than 55 µm for which Psm shows close agreement can be explained by the fact that the polynomial fits described in section 4 may locally present large differences in order to reach a best fit for the whole range. Donovan’s ratios were retrieved from one single midlatitude location and involve uncertainties linked to the crystals’ habit, but the differences in the RF calculated with the bimodal model suggest that the inclusion of basic information like the Psm in the characterization of cirrus can improve the parametrization of their radiative properties.

8.

Conclusions

The uncertainties linked to the radiative modelling of cirrus clouds are related to the complex shapes of their particles and the retrieval of small crystals. The large range of size parameters observed in cirrus and the lack of a natural coordinate system in which Maxwell’s equations can be naturally solved have limited the use of realistic shapes and exact numerical techniques. T-Matrix single scattering properties of hexagonal cylinders integrated over a set of lidar–radar retrievals and over a compilation of in situ-measured distributions were used to assess the differences in the calculated radiative forcing due to the shape of the size distributions used in producing the parametrization. The discrepancies reached 10 W m−2 , evidencing the ambiguity linked to the two-parameter representation and the need to include information about the shape of the distributions in the characterization of cirrus. Our results suggest that the inclusion of simple information (like the relative concentration of small particles) provides a way to produce more meaningful parametrizations. More extensive and accurate retrievals of small particles in tropical and midlatitudes are needed before reliable parametrizations of the radiative properties of cirrus will be obtained. Acknowledgements The authors wish to thank Dr Anthony Baran for making available to us the database used in this study (Baran et al., 2002). We are grateful to Dr Qiang Fu for access to his size distribution compilation and to Dr Wenyi Zhong for helpful discussions. We appreciate the helpful suggestions and comments given by reviewers and the editor during revision of the manuscript. References Asano S, Yamamoto G. 1975. Light scattering by a spheroidal particle. Appl. Optics 14(1): 29–49. Auer AH Jr, Veal DL. 1970. The dimension of ice crystals in natural clouds. J. Atmos. Sci. 27: 919–926. Copyright  2007 Royal Meteorological Society

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Q. J. R. Meteorol. Soc. 133: 273–282 (2007) DOI: 10.1002/qj