SPECTRAL OPTICAL PROPERTIES OF PURE ICE AND SNOW
Following from: Solar heating and melting of snowpacks and ice sheets in polar regions
1. OPTICAL CONSTANTS OF ICE IN THE VISIBLE AND NEAR-INFRARED
Two spectral optical constants are usually considered as real and imaginary parts of the complex index of refraction, m(λ) = n(λ) – iκ(λ), where n is the index of refraction and κ is the index of absorption (Born and Wolf, 1999). Spectral behavior of the indices of refraction and absorption are not independent of each other but satisfy the Kramers–Krönig relation (Lucarini et al., 2005). Particularly, the index of refraction is almost constant in the spectral ranges of a very low absorption, such as that for water ice in the visible range. The spectral optical constants of ice obtained by Warren and Brandt (2008) are plotted in Fig. 1. The wavelength range shown in Fig. 1 is important not only for the radiative heating problem under consideration but also for some problems related with the ultraviolet radiation.
(a) | (b) |
Figure 1. Spectral optical constants of ice: (a) index of refraction and (b) index of absorption
The extremely low value of κ in the visible and a significant increase of the absorption index in the near-infrared range determine the specific spectral properties of ice grains and snow in these spectral ranges. In particular, the known high value of snow albedo is a result of almost perfect spectral transparency of pure ice (Wiscombe and Warren, 1980; Warren, 1982; Kokhanovsky and Zege, 2004). Obviously, even very small impurities, such as atmospheric aerosol particles, may significantly increase the value of κ, and this strongly affects the observed albedo of snow. One can also recommend several recent papers on the specific problem of snow albedo: Gardner and Sharp (2010), Brandt et al. (2011), Malinka et al. (2016), Wang et al. (2017), Kokhanovsky et al. (2018), Picard et al. (2020), and He and Flanner (2020).
2. OPTICAL PROPERTIES OF SNOW
Theoretical modeling of optical properties of various turbid media is often based on the classical Mie theory for spherical particles (Van de Hulst, 1981; Bohren and Huffman, 1983; Hergert and Wriedt, 2012). The size of the snow particles is much greater that the wavelength of solar radiation, and the shape of these particles may be very complex. Therefore, the geometrical optics (GO) approximation and some other advanced methods are used to calculate optical properties of single particles and the resulting properties of snow (Bi et al., 2011; Borovoi et al., 2014; Lindqvist et al., 2018). Alternative approaches for optical properties of ice grains of complex shape can be found in papers (Grenfell and Warren, 1999; Neshyba et al., 2003; Libois et al., 2013; Räisänen et al., 2015; Ishimoto et al., 2018). However, for the sake of simplicity, one should not forget about the physically sound analytical approximations suggested in early papers by Irvine (1963, 1964), Kokhanovsky and Zege (1995, 1996), Dombrovsky (1996), and Dombrovsky and Baillis (2010).
In this article, the spherical ice grains of different sizes are considered instead of ice particles of complex shape and orientation. The calculations are performed for ice grains of radius a = 50, 100, and 200 μm. These variants can be treated as those corresponding to different snow morphology. The case of nonspherical particles can be treated as well. However, this study is focused on snowpack heating, and the problem of nonsphericity of ice grains is of secondary importance.
The most general solution for the optical properties of homogeneous spherical particles is given by the rigorous Mie theory. However, the ice grains considered are much greater in size than the wavelength. This makes it reasonable to consider the GO approximation as the main tool in this work. At the same time, the Mie theory, with the use of computer code published in Dombrovsky (1996) and described also by Dombrovsky and Baillis (2010), is used for the reference calculations.
With the use of transport approximation, only two dimensionless characteristics of absorption and scattering of single particles are necessary: the absorption efficiency factor, Q_{a}, and the transport efficiency factor of scattering, Q_{s}^{tr} = Q_{s} × (1 – μ¯ ), where μ¯ is the asymmetry factor of scattering. The values of transport efficiency factor of extinction, Q_{tr} = Q_{a} + Q_{s}^{tr}, and transport albedo of the particle, ω_{tr} = Q_{s}^{tr}/Q_{tr} are also considered. The mentioned characteristics depend on spectral optical constants and also on the diffraction parameter, x = 2πa/λ, introduced in the Mie theory.
The results of calculations using the GO solution obtained by Kokhanovsky and Zege (1995) are compared to calculations based on the Mie theory in Fig. 2. One can see that GO can be used to calculate both Q_{a} and ω_{tr} for ice grains of various size. As a result, the important value of transport extinction coefficient, Q_{tr} = Q_{a} /(1 – ω_{tr}), can be also obtained using the GO solution for the case of κ ≪ n:
(1a) |
(1b) |
where
(1c) |
(1d) |
(1e) |
(a) | (b) |
Figure 2. Optical properties of ice grains: (a) efficiency factor of absorption and (b) transport albedo
To avoid numerical errors of direct calculations by Eqs. (1a)–(1e) at Q_{a} < 10^{–5}, one can use the following approximation, which is correct in the case of κx ≪ 1:
(2) |
An approximation suggested by Dombrovsky (2002) for arbitrary values of x is also appropriate.
One can see in Fig. 2(b) that light scattering highly predominates very weak absorption in the visible spectral range. This results in a strong reflection of visible solar radiation from the snow surface and relatively small contribution of this spectral range to the heating of snow surface layer. At the same time, the remaining (not reflected) visible part of the collimated solar radiation forms almost diffuse the radiation field in the snow. As a result, the visible light is absorbed relatively far from the snow surface and its contribution to heating deep layers of snow is considerable. On the contrary, the reflection of near-infrared solar radiation from the snow surface is relatively small and this spectral range contributes strongly to the radiation power absorbed in the surface layer.
It is assumed that the hypothesis of independent scattering by single ice grains in snow is true (Mishchenko, 2018). It means that each particle absorbs and scatters the radiation in exactly the same manner as if other particles do not exist. In addition, there is no systematic phase relation between partial waves scattered by individual particles during the observation time interval, so that the intensities of the partial waves can be added without regard to phase. In other words, each particle is in the far-field zones of all other particles and scattering by individual particles is incoherent.
REFERENCES
Bi, L., Yang, P., Kattawar, G.W., Hu, Y., and Baum, B.A. (2011) Scattering and Absorption of Light by Ice Particles: Solution by a New Physical-Geometrical Optics Hybrid Method, J. Quant. Spectrosc. Radiat. Transf., 112(9): 1492–1508.
Bohren, C.F. and Huffman, D.R. (1983) Absorption and Scattering of Light by Small Particles, New York: Wiley.
Born, M. and Wolf E. (1999) Principles of Optics, 7th (expanded) ed., New York: Cambridge University Press.
Borovoi, A., Konoshonkin, A., and Kustova, N. (2014) The Physical-Optics Approximation and Its Application to Light Backscattering by Hexagonal Ice Crystals, J. Quant. Spectrosc. Radiat. Transf., 146: 181–189.
Brandt, R.E., Warren, S.G., and Clarke A.D. (2011) A Controlled Snow-Making Experiment Testing the Relation Between Black Carbon Content and Reduction of Snow Albedo, J. Geophys. Res. Atmos., 116(D8): D08109.
Dombrovsky, L.A. (1996) Radiation Heat Transfer in Disperse Systems, Danbury, CT: Begell House.
Dombrovsky, L.A. (2002) Spectral Model of Absorption and Scattering of Thermal Radiation by Droplets of Diesel Fuel, High Temp., 40(2): 242–248.
Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, Danbury, CT: Begell House.
Gardner, A.S. and Sharp, M.J. (2010) A Review of Snow and Ice Albedo and the Development of a New Physically Based Broadband Albedo Parameterization, J. Geophys. Res. Earth. Surf., 115(F1): F01009.
Grenfell, T. and Warren, S. (1999) Representation of a Nonspherical Ice Particle by a Collection of Independent Spheres for Scattering and Absorption of Radiation, J. Geophys. Res. Atmos., 104(D24): 31697–31709.
He, C. and Flanner, M. (2020) Snow Albedo and Radiative Transfer: Theory, Modeling, and Parametrization, in Kokhanovsky, A. (Ed.), Springer Series in Light Scattering, Vol. 5, Cham, Switzerland: Springer, 67–133.
Hergert, W. and Wriedt, T. (2012) The Mie Theory: Basics and Applications, Berlin: Springer.
Irvine, W.M. (1963) The Asymmetry of the Scattering Diagram of a Spherical Particle, Commun. Observ. Leiden., 17(3): 176–184.
Irvine, W.M. (1964) Light Scattering by Spherical Particles: Radiation Pressure, Asymmetry Factor, and Extinction Cross Section, J. Opt. Soc. Am., 55(1): 16–21.
Ishimoto, H., Adachi, S., Yamaguchi, S., Tanikawa, T., Aoki, T., and Masuda, K. (2018) Snow Particles Extracted from X-Ray Computed Microtomography Imagery and Their Single-Scattering Properties, J. Quant. Spectrosc. Radiat. Transf., 209: 113–128.
Kokhanovsky, A.A. and Zege, E.P. (2004) Scattering Optics of Snow, Appl. Opt., 43(7): 1589–1602.
Kokhanovsky, A.A. and Zege, E.P. (1995) Local Optical Parameters of Spherical Polydispersions: Simple Approximations, Appl. Opt., 34(24): 5513–5519.
Kokhanovsky, A.A. and Zege, E.P. (1996) Optical Properties of Aerosol Particles: A Review of Approximate Analytical Solutions, J. Aerosol Sci., 28(1): 1–21.
Kokhanovsky, A., Lamare, M., Di Mauro, B., Picard, G., Arnaud, L., Dumont, M., Tuzet, F., Brockmann, C., and Box, J.E. (2018) On the Reflectance Spectroscopy of Snow, Cryosphere, 12(7): 2371–2382.
Libois, Q., Picard, G., France, J.L., Arnaud, L., Dumont, M., Carmagnola, C.M., and King, M.D. (2013) Influence of Grain Shape on Light Penetration in Snow, Cryosphere, 7(6): 1803–1818.
Lindqvist, H., Martikainen, J., Räbinä, J., Penttilä, A., and Muinonen, K. (2018) Ray Optics for Absorbing Particles with Application to Ice Crystals at Near-Infrared Wavelengths, J. Quant. Spectrosc. Radiat. Transf., 217: 329–337.
Lucarini, V., Saarinene, J.J., Peiponen, K.-E., and Vartainen, E.M. (2005) Kramers–Krönig Relations in Optical Material Research, Berlin: Springer.
Malinka, A., Zege, E., Heygster, G., and Istomina, L. (2016) Reflective Properties of White Sea Ice and Snow, Cryosphere, 10(6): 2541–2557.
Mishchenko, M.I. (2018) “Independent” and “Dependent” Scattering by Particles in a Multi-Particle Group, OSA Continuum, 1(1): 243–260.
Neshyba, S.P., Grenfell, T.C., and Warren, S.G. (2003) Representation of a Nonspherical Ice Particle by a Collection of Independent Spheres for Scattering and Absorption of Radiation: 2. Hexagonal Columns and Plates, J. Geophys. Res. Atmos., 108(D15): Article ID 4448.
Picard, G., Dumont, M., Lamare, M., Tuzet, F., Larue, F., Pirazzini, R., and Arnaud, L. (2020) Spectral Albedo Measurements Over Snow-Covered Slopes: Theory and Slope Effect Corrections, Cryosphere, 14(5): 1497–1517.
Räisänen, P., Kokhanovsky, A., Guyot, G., Jourdan, O., and Nousiainen, T. (2015) Parametrization of Single-Scattering Properties of Snow, Cryosphere, 9(3): 1277–1301.
Van de Hulst, H.C. (1981) Light Scattering by Small Particles, New York: Dover.
Wang, X., Pu, W., Ren, Y., Zhang, X., Zhang, X., Shi, J., Jin, H., Dai, M., and Chen, Q. (2017) Observations and Model Simulations of Snow Albedo Reduction in Seasonal Snow Due to Insoluble Light-Absorbing Particles During 2014 Chinese Survey, Atmos. Chem. Phys., 17(3): 2279–2296.
Warren, S.G. (1982) Optical Properties of Snow, Rev. Geophys. Space Phys., 20(1): 67–89.
Warren, S.G. and Brandt, R.E. (2008) Optical Constants of Ice from the Ultraviolet to the Microwave: A Revised Compilation, J. Geophys. Res. Atmos., 113(D14): D14220.
Wiscombe, W.J. and Warren, S.G. (1980) A Model for the Spectral Albedo of Snow, I: Pure Snow, J. Atmos. Sci., 37(12): 2712–2733.
References
- Bi, L., Yang, P., Kattawar, G.W., Hu, Y., and Baum, B.A. (2011) Scattering and Absorption of Light by Ice Particles: Solution by a New Physical-Geometrical Optics Hybrid Method, J. Quant. Spectrosc. Radiat. Transf., 112(9): 1492–1508.
- Bohren, C.F. and Huffman, D.R. (1983) Absorption and Scattering of Light by Small Particles, New York: Wiley.
- Born, M. and Wolf E. (1999) Principles of Optics, 7th (expanded) ed., New York: Cambridge University Press.
- Borovoi, A., Konoshonkin, A., and Kustova, N. (2014) The Physical-Optics Approximation and Its Application to Light Backscattering by Hexagonal Ice Crystals, J. Quant. Spectrosc. Radiat. Transf., 146: 181–189.
- Brandt, R.E., Warren, S.G., and Clarke A.D. (2011) A Controlled Snow-Making Experiment Testing the Relation Between Black Carbon Content and Reduction of Snow Albedo, J. Geophys. Res. Atmos., 116(D8): D08109.
- Dombrovsky, L.A. (1996) Radiation Heat Transfer in Disperse Systems, Danbury, CT: Begell House.
- Dombrovsky, L.A. (2002) Spectral Model of Absorption and Scattering of Thermal Radiation by Droplets of Diesel Fuel, High Temp., 40(2): 242–248.
- Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, Danbury, CT: Begell House.
- Gardner, A.S. and Sharp, M.J. (2010) A Review of Snow and Ice Albedo and the Development of a New Physically Based Broadband Albedo Parameterization, J. Geophys. Res. Earth. Surf., 115(F1): F01009.
- Grenfell, T. and Warren, S. (1999) Representation of a Nonspherical Ice Particle by a Collection of Independent Spheres for Scattering and Absorption of Radiation, J. Geophys. Res. Atmos., 104(D24): 31697–31709.
- He, C. and Flanner, M. (2020) Snow Albedo and Radiative Transfer: Theory, Modeling, and Parametrization, in Kokhanovsky, A. (Ed.), Springer Series in Light Scattering, Vol. 5, Cham, Switzerland: Springer, 67–133.
- Hergert, W. and Wriedt, T. (2012) The Mie Theory: Basics and Applications, Berlin: Springer.
- Irvine, W.M. (1963) The Asymmetry of the Scattering Diagram of a Spherical Particle, Commun. Observ. Leiden., 17(3): 176–184.
- Irvine, W.M. (1964) Light Scattering by Spherical Particles: Radiation Pressure, Asymmetry Factor, and Extinction Cross Section, J. Opt. Soc. Am., 55(1): 16–21.
- Ishimoto, H., Adachi, S., Yamaguchi, S., Tanikawa, T., Aoki, T., and Masuda, K. (2018) Snow Particles Extracted from X-Ray Computed Microtomography Imagery and Their Single-Scattering Properties, J. Quant. Spectrosc. Radiat. Transf., 209: 113–128.
- Kokhanovsky, A.A. and Zege, E.P. (2004) Scattering Optics of Snow, Appl. Opt., 43(7): 1589–1602.
- Kokhanovsky, A.A. and Zege, E.P. (1995) Local Optical Parameters of Spherical Polydispersions: Simple Approximations, Appl. Opt., 34(24): 5513–5519.
- Kokhanovsky, A.A. and Zege, E.P. (1996) Optical Properties of Aerosol Particles: A Review of Approximate Analytical Solutions, J. Aerosol Sci., 28(1): 1–21.
- Kokhanovsky, A., Lamare, M., Di Mauro, B., Picard, G., Arnaud, L., Dumont, M., Tuzet, F., Brockmann, C., and Box, J.E. (2018) On the Reflectance Spectroscopy of Snow, Cryosphere, 12(7): 2371–2382.
- Libois, Q., Picard, G., France, J.L., Arnaud, L., Dumont, M., Carmagnola, C.M., and King, M.D. (2013) Influence of Grain Shape on Light Penetration in Snow, Cryosphere, 7(6): 1803–1818.
- Lindqvist, H., Martikainen, J., Räbinä, J., Penttilä, A., and Muinonen, K. (2018) Ray Optics for Absorbing Particles with Application to Ice Crystals at Near-Infrared Wavelengths, J. Quant. Spectrosc. Radiat. Transf., 217: 329–337.
- Lucarini, V., Saarinene, J.J., Peiponen, K.-E., and Vartainen, E.M. (2005) Kramers–Krönig Relations in Optical Material Research, Berlin: Springer.
- Malinka, A., Zege, E., Heygster, G., and Istomina, L. (2016) Reflective Properties of White Sea Ice and Snow, Cryosphere, 10(6): 2541–2557.
- Mishchenko, M.I. (2018) “Independent” and “Dependent” Scattering by Particles in a Multi-Particle Group, OSA Continuum, 1(1): 243–260.
- Neshyba, S.P., Grenfell, T.C., and Warren, S.G. (2003) Representation of a Nonspherical Ice Particle by a Collection of Independent Spheres for Scattering and Absorption of Radiation: 2. Hexagonal Columns and Plates, J. Geophys. Res. Atmos., 108(D15): Article ID 4448.
- Picard, G., Dumont, M., Lamare, M., Tuzet, F., Larue, F., Pirazzini, R., and Arnaud, L. (2020) Spectral Albedo Measurements Over Snow-Covered Slopes: Theory and Slope Effect Corrections, Cryosphere, 14(5): 1497–1517.
- Räisänen, P., Kokhanovsky, A., Guyot, G., Jourdan, O., and Nousiainen, T. (2015) Parametrization of Single-Scattering Properties of Snow, Cryosphere, 9(3): 1277–1301.
- Van de Hulst, H.C. (1981) Light Scattering by Small Particles, New York: Dover.
- Wang, X., Pu, W., Ren, Y., Zhang, X., Zhang, X., Shi, J., Jin, H., Dai, M., and Chen, Q. (2017) Observations and Model Simulations of Snow Albedo Reduction in Seasonal Snow Due to Insoluble Light-Absorbing Particles During 2014 Chinese Survey, Atmos. Chem. Phys., 17(3): 2279–2296.
- Warren, S.G. (1982) Optical Properties of Snow, Rev. Geophys. Space Phys., 20(1): 67–89.
- Warren, S.G. and Brandt, R.E. (2008) Optical Constants of Ice from the Ultraviolet to the Microwave: A Revised Compilation, J. Geophys. Res. Atmos., 113(D14): D14220.
- Wiscombe, W.J. and Warren, S.G. (1980) A Model for the Spectral Albedo of Snow, I: Pure Snow, J. Atmos. Sci., 37(12): 2712–2733.
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