Water Droplets and Bubbles in Microwave Spectral Range
Following from: The Mie solution for spherical particles
Leading to: Thermal microwave radiation of disperse systems on the sea surface
The propagation of radio waves in the atmosphere is one of the most important problems of radiophysics due to its application in radio communications, as well as in microwave remote sensing of the atmosphere and the earth’s surface (Basharinov et al., 1974; Armand et al., 1977; Dombrovsky, 1981; Oguchi, 1983; Dombrovsky and Raizer, 1992; Cherny and Raizer, 1998; Yueh et al., 1999; Piepmeier and Gasiewski, 2001; Rose et al., 2002; Sharkov, 2003, 2007; Camps et al., 2005; Raizer, 2005, 2006, 2007; Padmanabhan et al., 2006). Various hydrometeors in the atmosphere (water droplets, ice particles, etc.) significantly affect electromagnetic wave propagation in the millimeter and centimeter ranges (Oguchi, 1983). For this reason, the absorption and scattering of electromagnetic waves by these particles (first of all, by water droplets) have attracted the attention of many researchers for a long time. The passive remote sensing of the ocean surface from aircraft deals with the microwave characteristics of small water droplets in water sprays produced by wind wave breaking (Cherny and Raizer, 1998; Sharkov, 2003, 2007). Thermal radio emission from foam structures appears to be especially important because the foam on the ocean surface increases the emissivity and brightness temperature measured by a passive microwave radiometer and is a key component of the wind speed signal. Given certain assumptions, one can obtain the emission characteristics of the foam structures on the basis of the analysis of the microwave properties of single water bubbles (Dombrovsky, 1979; Dombrovsky and Raizer, 1992; Camps et al., 2005; Raizer, 2005, 2006, 2007).
Spherical water droplets and thin-wall bubbles are considered in this section. The properties of these particles in the millimeter and centimeter wavelength ranges are of interest in scattering theory, and for practical problems of microwave remote sensing of the atmosphere and ocean surface. Here, we follow the early work of Dombrovsky (1979, 1981) and Dombrovsky and Raizer (1992), with some additional numerical data in this section.
It is important to understand the spectral dependences of the complex index of refraction of water in order to calculate the microwave properties of water droplets and bubbles. The optical properties of fresh water have been investigated in great detail over the whole spectrum: not only in the visible and infrared (Rusk and Williams, 1971; Hale and Querry, 1973), but also in the radio wave range (Ray, 1972; Zolotarev and Dyomin, 1977). Data for salt sea water have been reported by Stogryn (1971) and Raizer et al. (1975). Following the work of Dombrovsky (1979, 1981), the complex permittivity of water is determined by the Debye’s relaxation model by taking into account the ionic conductivity:
where λs is the relaxation wavelength; ε∞ and εs are the high- and low-frequency permittivities, respectively; and σ is the electric conductivity. The values of dielectric parameters λs, ε∞, εs, and σ depend on water salinity and temperature. The corresponding expressions given by Stogryn (1971) are
and N is the normality of the NaCl solution:
In Eqs. (1)-(5), salinity s is measured in parts per thousand (ppt, pro mil, ‰), temperature t is measured in centigrade degrees, and electric conductivity σ is measured in (Ohm ·mm)-1. The spectral dependences of the indices of refraction and absorption calculated by Eqs. (1)-(5) and formula
are shown in Fig. 1. When the wavelength increases from 2 to 80 mm the index of refraction increases monotonically up to the values typical of metals in the infrared. The index of absorption has a maximum in the range of λ = 10 - 20 mm and does exceed the value κ = 3. The temperature increase leads to an increase in n and displacement of the maximum of κ(λ) to the shorter waves. Both the values of n and κ for salt water differ slightly from the corresponding values for fresh water at wavelengths of less than 20 mm. At the same time, the index of absorption for salt water in the long-wave region is considerably greater than that for fresh water (the refractive index is somewhat less than for fresh water). The last circumstance, as will be shown below, affects the radiative properties of large water droplets at λ > 50 mm to a considerable degree.
Figure 1. Optical constants of fresh and salt water: 1 and 2, fresh water (s = 0), and 3 and 4, salt water at s = 35‰; 1 and 3, T = 283 K, and 2 and 4, T = 293 K.
The computational data reported by Deirmendjian (1969), Shifrin and Chernyak (1967), and Ishimaru (1978) for the radiative properties of single water droplets are not complete in the spectral range under consideration. Some additional results of Mie theory calculations can be found in the research done by Dombrovsky and Raizer (1992). The microwave properties of typical water droplets produced by wind-wave breaking (Bortkovsky and Timanovsky, 1982; Bezzabotnov et al., 1986) calculated by the Mie theory are presented in Fig. 2. For small droplets of radius a = 0.5 mm, Rayleigh scattering takes place at λ > 6 mm, where Qa >> Qs, |μ| << 1. The Rayleigh scattering region becomes narrower for larger droplets. At a = 4 mm, this regime is not realized at all in the spectral range under consideration. The asymmetry factor of scattering is negative near the Rayleigh region due to large values of n; i.e., backward scattering is predominant, as in small metal particles in the infrared. All of the special features of the spectral dependences of Qa, Qstr, and μ remain the same in the temperature interval of 0-20°C due to the monotonic temperature dependence of the water dielectric parameters. One can see in Fig. 2 that the effect of water salinity on the radiative properties of water droplets is significant only for large droplets in the centimeter wavelength range. For fresh water droplets of radius a = 4 mm, we have found the resonance absorption of the microwave radiation at λ ≈ 70 mm with Qa ≈ 1.7 in the maximum. The same effect for droplets of salt water at s = 35‰ is not significant.
Figure 2. Spectral radiative properties of water droplets at temperature T = 293 K: solid curves, fresh water (s = 0); dashed curves, salt water at s = 35‰.
Consider the microwave properties of a spherical water bubble with a radius in the range corresponding to the disperse structure of sea foam (Raizer et al., 1976; Raizer and Sharkov, 1980; Peltzer and Griffin, 1988). Some computational results for water bubbles at five wavelengths from 2.6 mm up to 180 mm were given by Dombrovsky (1979, 1981, 1996) and also by Dombrovsky and Baillis (2010). In connection with microwave scattering by water bubbles, it is also worth noting the early study by Battan and Herman (1961), where some backscattering data at one wavelength, λ = 32.1 mm, in a more wide range of the bubble radius are presented. In this section, we give more detailed data over the spectrum for thin-wall water bubbles. The calculated spectral dependences Qa(λ), Qstr(λ), and μ(λ) for fresh water bubbles of various external radii a at water shell thicknesses δ1 = 5 and 10 μm are shown in Fig. 3. One can see that water bubbles scatter radiation very weakly: Qstr << Qa. At the same time, the scattering in the short-wave range is anisotropic and differs significantly from the Rayleigh scattering.
Figure 3. Spectral radiative properties of fresh water bubbles at temperature T = 293 K: solid curves, δ1 = 5 μm; dashed curves, δ1 = 10 μm.
The variation of the efficiency factor of absorption with the bubble radius at the wavelength λ = 2.6 mm is shown in Fig. 4. One can see that the Rayleigh theory is applicable for the absorption calculations in bubbles with a radius of less than 1 mm. The Rayleigh formula can be additionally simplified taking into account the low relative thickness of the bubble wall:
It is of interest that Eq. (7) differs from the limiting expression for δ1 → 0. In this aspect, the water bubbles considered are not thin-wall particles.
Figure 4. Absorption efficiency factor of fresh water bubbles at wavelength λ = 2.6 mm and temperature T = 293 K: 1, Mie theory; 2, Rayleigh theory; 3, approximate formula (7).
In contrast to absorption, the scattering characteristics of water bubbles are not described by Rayleigh scattering theory. The details of microwave scattering by water bubbles can be found in the study done by Dombrovsky (1981). It is important to note that even comparatively large bubbles with a thin wall are characterized by very weak scattering of microwave radiation. This result enables us to explain the experimental results of high microwave emissivity of foam structures on the sea surface (Bordonskii et al., 1978; Dombrovsky, 1979). At the same time, the demonstrated applicability of the electric-dipole model for microwave absorption by a single bubble is the basis of the approximate theoretical models for thermal radio-emission of foam-like structures.
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