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Abstract
Effective-medium theories yield effective dielectric functions (or, equivalently, refractive indices) of composite media. Such theories have been formulated that go beyond the Maxwell-Garnett and Bruggeman theories, which art restricted to media composed of grains much smaller than the wavelength. These extended effective-medium theories do not, however, yield effective dielectric functions that can be used for the same purposes for which we unhesitatingly use the dielectric functions of substances such as pure water and pure ice (e.g., reflection and transmission by smooth interfaces; absorption and scattering by particles). Extended dielectric functions can lead to unphysical results; for example, absorption in composite media with nonabsorbing components. Moreover, if the grains in composite media are large enough to give rise to magnetic dipole and higher-order multipole radiation, then the effective permeability of the composite medium cannot be taken to be that of free space even if the grains are nonmagnetic.
Recently, extended effective-medium theories have been applied to the problem of determining, the effective dielectric function of ice in which soot grains are embedded in order to explain a factor of 2 discrepancy between measurements of the albedo of soot-contaminated snow and calculations based on a snow albedo model. Setting aside questions about the applicability of these theories, reasonable alternative explanations for the discrepancy exist: (i) Soot is not an invariable substance; measured refractive indices of carbonaceous materials vary appreciably, especially the imaginary part (about a factor of 5). (ii) Absorption by a small soot particle depends on its shape, varying by as much as a factor of 2. (iii) Absorption by a soot particle may be enhanced by porosity; for a fixed particle volume, the enhancement is roughly proportional to the porosity. To predict exactly how much a given amount of soot reduces the visible albedo of snow requires, therefore, more detailed information about the soot than is likely to be readily obtainable.
Abstract
Effective-medium theories yield effective dielectric functions (or, equivalently, refractive indices) of composite media. Such theories have been formulated that go beyond the Maxwell-Garnett and Bruggeman theories, which art restricted to media composed of grains much smaller than the wavelength. These extended effective-medium theories do not, however, yield effective dielectric functions that can be used for the same purposes for which we unhesitatingly use the dielectric functions of substances such as pure water and pure ice (e.g., reflection and transmission by smooth interfaces; absorption and scattering by particles). Extended dielectric functions can lead to unphysical results; for example, absorption in composite media with nonabsorbing components. Moreover, if the grains in composite media are large enough to give rise to magnetic dipole and higher-order multipole radiation, then the effective permeability of the composite medium cannot be taken to be that of free space even if the grains are nonmagnetic.
Recently, extended effective-medium theories have been applied to the problem of determining, the effective dielectric function of ice in which soot grains are embedded in order to explain a factor of 2 discrepancy between measurements of the albedo of soot-contaminated snow and calculations based on a snow albedo model. Setting aside questions about the applicability of these theories, reasonable alternative explanations for the discrepancy exist: (i) Soot is not an invariable substance; measured refractive indices of carbonaceous materials vary appreciably, especially the imaginary part (about a factor of 5). (ii) Absorption by a small soot particle depends on its shape, varying by as much as a factor of 2. (iii) Absorption by a soot particle may be enhanced by porosity; for a fixed particle volume, the enhancement is roughly proportional to the porosity. To predict exactly how much a given amount of soot reduces the visible albedo of snow requires, therefore, more detailed information about the soot than is likely to be readily obtainable.
Abstract
The visual phenomenon called virga, a sudden change in the brightness of a precipitation shaft below a cloud, is commonly attributed to evaporation of raindrops. It is said to be rain that does not reach the ground. The optical thickness of an evaporating rain shaft, however, decreases gradually from cloud base to ground. Thus, it is more likely that virga results from snowflakes melting in descent. Horizontal optical-thickness decreases of more than ten can occur in the short distance over which a snowflake is transformed into a raindrop. This decrease is caused by two factors: a smaller number density of hydrometeors because of the greater fall velocity of raindrops than of equivolume snowflakes, and a smaller scattering cross section: the first of these is dominant. An alternative explanation of virga is that it is precipitation that has not yet reached (rather than does not reach) the ground. This is a plausible explanation given the long time periods it may take hydrometeors, especially snowflakes, to descend from cloud to ground.
Abstract
The visual phenomenon called virga, a sudden change in the brightness of a precipitation shaft below a cloud, is commonly attributed to evaporation of raindrops. It is said to be rain that does not reach the ground. The optical thickness of an evaporating rain shaft, however, decreases gradually from cloud base to ground. Thus, it is more likely that virga results from snowflakes melting in descent. Horizontal optical-thickness decreases of more than ten can occur in the short distance over which a snowflake is transformed into a raindrop. This decrease is caused by two factors: a smaller number density of hydrometeors because of the greater fall velocity of raindrops than of equivolume snowflakes, and a smaller scattering cross section: the first of these is dominant. An alternative explanation of virga is that it is precipitation that has not yet reached (rather than does not reach) the ground. This is a plausible explanation given the long time periods it may take hydrometeors, especially snowflakes, to descend from cloud to ground.
Abstract
Shape can strongly affect scattering by particles at wavelengths near bulk absorption bands. An example is scattering by many components of the atmospheric aerosol at wavelengths in the transmission window centered around 10 μm. An exact solution to the problem of scattering by an irregular particle, even if it were available, would yield more detail than necessary. An alternative approach is to formulate the irregular particle scattering problem in terms of the kinds of averages inherent in the optical properties of an ensemble of particles. Such a statistical method has been applied to an ensemble of small irregular particles by averaging over a range of electromagnetic microstates, in this instance randomly oriented anisotropic oscillators (i.e., Rayleigh ellipsoids). Infrared backscattering spectra calculated by this method agree better with laboratory-measured spectra for ammonium sulfate particles than those calculated using Mie theory.
Abstract
Shape can strongly affect scattering by particles at wavelengths near bulk absorption bands. An example is scattering by many components of the atmospheric aerosol at wavelengths in the transmission window centered around 10 μm. An exact solution to the problem of scattering by an irregular particle, even if it were available, would yield more detail than necessary. An alternative approach is to formulate the irregular particle scattering problem in terms of the kinds of averages inherent in the optical properties of an ensemble of particles. Such a statistical method has been applied to an ensemble of small irregular particles by averaging over a range of electromagnetic microstates, in this instance randomly oriented anisotropic oscillators (i.e., Rayleigh ellipsoids). Infrared backscattering spectra calculated by this method agree better with laboratory-measured spectra for ammonium sulfate particles than those calculated using Mie theory.
Abstract
Calculations have been made of the radar backscattering and attenuation cross sections of dry and spongy ice spheres. One set of calculations was the cross sections of spheres with diameters exponentially distributed. As expected, attenuation cross sections are greater at a wavelength of 3.21 cm than at 5.05 and 10.0 cm. Calculations were also made of attenuation by monodisperse distributions of spheres composed of spongy ice and having diameters as large as about 8 cm. Attenuation of 3-cm radiation by dry ice and spongy ice spheres can be very large. At most diameters and water volume fractions, the one-way attenuation of 10-cm radiation by monodisperse spheres, in concentrations giving a radar reflectivity of 60 dBZ, is negligibly small (i.e., <0.1 dB km−1), but at a few diameters and water fractions, attenuation can be substantially larger. Although in most circumstances attenuation increases as wavelength decreases, there are exceptions at some diameters and water volume fractions. These calculations may explain observations that C-band attenuation in hailstorms is not always larger than S-band attenuation.
Abstract
Calculations have been made of the radar backscattering and attenuation cross sections of dry and spongy ice spheres. One set of calculations was the cross sections of spheres with diameters exponentially distributed. As expected, attenuation cross sections are greater at a wavelength of 3.21 cm than at 5.05 and 10.0 cm. Calculations were also made of attenuation by monodisperse distributions of spheres composed of spongy ice and having diameters as large as about 8 cm. Attenuation of 3-cm radiation by dry ice and spongy ice spheres can be very large. At most diameters and water volume fractions, the one-way attenuation of 10-cm radiation by monodisperse spheres, in concentrations giving a radar reflectivity of 60 dBZ, is negligibly small (i.e., <0.1 dB km−1), but at a few diameters and water fractions, attenuation can be substantially larger. Although in most circumstances attenuation increases as wavelength decreases, there are exceptions at some diameters and water volume fractions. These calculations may explain observations that C-band attenuation in hailstorms is not always larger than S-band attenuation.
Abstract
Calculations of radar backscattering by inhomogeneous precipitation particles require values of the dielectric function of two-component mixtures. Four such dielectric functions are critically examined and their relative merits are weighed. Although apparently different, two are shown to be equivalent: the effective-medium and Polder-van Santen theories. All the dielectric functions agree when the two components are dielectrically similar. All except the Maxwell-Garnet dielectric function are symmetric with respect to interchange of the components. When compared with measurements on ice-air mixtures, the effective-medium and Maxwell-Garnet dielectric functions are marginally better than the Debye function, which has previously been used in backscattering calculations. When the fraction of water is high, the effective-medium function gives calculated values of radar backscattering that are in good agreement with measurements on ice spheres coated with a mixture of ice and water. The Maxwell-Garnet theory, with ice inclusions embedded in a water matrix, is also in good agreement with these measurements, and is so over a wider range of water-volume fractions than the effective-medium theory. Although there are no compelling reasons for preferring one above the other, on the basis of the evidence presented, we would be inclined to use the Maxwell-Garnet dielectric function in radar backscattering calculations.
Abstract
Calculations of radar backscattering by inhomogeneous precipitation particles require values of the dielectric function of two-component mixtures. Four such dielectric functions are critically examined and their relative merits are weighed. Although apparently different, two are shown to be equivalent: the effective-medium and Polder-van Santen theories. All the dielectric functions agree when the two components are dielectrically similar. All except the Maxwell-Garnet dielectric function are symmetric with respect to interchange of the components. When compared with measurements on ice-air mixtures, the effective-medium and Maxwell-Garnet dielectric functions are marginally better than the Debye function, which has previously been used in backscattering calculations. When the fraction of water is high, the effective-medium function gives calculated values of radar backscattering that are in good agreement with measurements on ice spheres coated with a mixture of ice and water. The Maxwell-Garnet theory, with ice inclusions embedded in a water matrix, is also in good agreement with these measurements, and is so over a wider range of water-volume fractions than the effective-medium theory. Although there are no compelling reasons for preferring one above the other, on the basis of the evidence presented, we would be inclined to use the Maxwell-Garnet dielectric function in radar backscattering calculations.
Abstract
The radar backscattering cross section of a spongy ice hailstone—a mixture of ice and liquid water—depends on its size, shape and dielectric function. There are two types of theories of the effective dielectric function of two-component mixtures: Maxwell-Garnet and Bruggeman theories. In the latter, the two components are treated symmetrically, whereas in the former they are not. We have generalized the Maxwell-Garnet expression, originally derived for spherical inclusions in an otherwise homogeneous matrix, to ellipsoidal inclusions. When this expression, with all ellipsoidal shapes equally probable, is used in calculations of radar backscattering by ice spheres coated with spongy ice, the results are in generally good agreement with measured cross sections. Agreement is better if the inclusions are ellipsoidal rather than spherical, but only slightly so. Shape considerations are less important than taking ice to be the inclusions and liquid water to be the matrix.
Abstract
The radar backscattering cross section of a spongy ice hailstone—a mixture of ice and liquid water—depends on its size, shape and dielectric function. There are two types of theories of the effective dielectric function of two-component mixtures: Maxwell-Garnet and Bruggeman theories. In the latter, the two components are treated symmetrically, whereas in the former they are not. We have generalized the Maxwell-Garnet expression, originally derived for spherical inclusions in an otherwise homogeneous matrix, to ellipsoidal inclusions. When this expression, with all ellipsoidal shapes equally probable, is used in calculations of radar backscattering by ice spheres coated with spongy ice, the results are in generally good agreement with measured cross sections. Agreement is better if the inclusions are ellipsoidal rather than spherical, but only slightly so. Shape considerations are less important than taking ice to be the inclusions and liquid water to be the matrix.
Abstract
The severest test of a theory of scattering by particles is how well it calculates scattering in the backward direction. The coupled-dipole method can be used for accurately calculating backscattering at 94 GHz by hexagonal ice crystals. Backscattering by columns is markedly different from that by plates, which indicates that it might be possible to infer size and shape distributions of ice crystals using recently developed millimeter wave radar.
Abstract
The severest test of a theory of scattering by particles is how well it calculates scattering in the backward direction. The coupled-dipole method can be used for accurately calculating backscattering at 94 GHz by hexagonal ice crystals. Backscattering by columns is markedly different from that by plates, which indicates that it might be possible to infer size and shape distributions of ice crystals using recently developed millimeter wave radar.
Green thunderstorms are observed occasionally, yet with one exception they have received no scientific attention, experimental or theoretical. Fraser suggested that thunderstorms themselves are not green but that a thick thunderstorm provides a dark backdrop for green airlight near sundown. Greenness is a consequence of reddened sunlight illuminating selective scatterers along the observer's line of sight. Bohren's alternative explanation is that green thunderstorms may be a consequence of the intrinsic blueness of clouds because of selective absorption by pure water, liquid or solid. Most clouds are so thin that the light transmitted by them is not markedly colored because of selective absorption. Only the most massive clouds—large both vertically and horizontally—are thick enough to shift the color of incident sunlight upon transmission. If that incident light is sunlight reddened at sundown, the transmitted light can be perceptually green. These two explanations do not exclude one another but allow for multiple causes, including those not yet identified.
Green thunderstorms are observed occasionally, yet with one exception they have received no scientific attention, experimental or theoretical. Fraser suggested that thunderstorms themselves are not green but that a thick thunderstorm provides a dark backdrop for green airlight near sundown. Greenness is a consequence of reddened sunlight illuminating selective scatterers along the observer's line of sight. Bohren's alternative explanation is that green thunderstorms may be a consequence of the intrinsic blueness of clouds because of selective absorption by pure water, liquid or solid. Most clouds are so thin that the light transmitted by them is not markedly colored because of selective absorption. Only the most massive clouds—large both vertically and horizontally—are thick enough to shift the color of incident sunlight upon transmission. If that incident light is sunlight reddened at sundown, the transmitted light can be perceptually green. These two explanations do not exclude one another but allow for multiple causes, including those not yet identified.
Abstract
Calculations have been made of the radar backscattering, differential reflectivity, and circular depolarization ratio of large oblate spheroids composed of spongy ice. Results are compatible with laboratory measurements by earlier investigators. As expected, scattering of 10-cm radiation depends to an important extent on the size, water content, and axial ratios of the spheroids. Observations of differential reflectivities close to zero in hailstorms can be explained, as was done by V. N. Bringi and his associates, as resulting from the irregular shapes and tumbling of hailstones. But such observations could also be explained by size-distributed oblate spheroids with equivalent diameters greater than about 3.5 cm, falling with vertical symmetry axes.
Abstract
Calculations have been made of the radar backscattering, differential reflectivity, and circular depolarization ratio of large oblate spheroids composed of spongy ice. Results are compatible with laboratory measurements by earlier investigators. As expected, scattering of 10-cm radiation depends to an important extent on the size, water content, and axial ratios of the spheroids. Observations of differential reflectivities close to zero in hailstorms can be explained, as was done by V. N. Bringi and his associates, as resulting from the irregular shapes and tumbling of hailstones. But such observations could also be explained by size-distributed oblate spheroids with equivalent diameters greater than about 3.5 cm, falling with vertical symmetry axes.
