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Morel (1973a) showed that if a dispersion of particles has a power-law PSD and if absorption of light by the particles is negligible, then the spectrum of the scattering coefficient (and of the attenuation coefficient, for a dispersion medium with negligible absorption of light) follows a power-law. The slope, γ, of that law is related to the power-law slope, sD, of the PSD according to:
| γ = sD - 3 | (1) |
with sD typically falling between 3 and 4 (see also Power-law PSD for natural dispersions).
The power-law for the spectrum of the attenuation coefficient: c = const λn -γ, where λn is a normalized wavelength, has been known in the atmospheric physics since 1929 as the Ångström law (for example, Heitzenberg and Charlson 1996). The relationship between the slope of the wavelength spectrum of the scattering coefficient of a dispersion and the slope of a power-law PSD of the dispersion, expressed by Eq. 1, was also found by Fournier and Forand (1994), Diehl and Haardt (1980), Volz (1954), and Junge (1952; as quoted by Heitzenberg and Charlson 1996).
Boss et al (2001), explored limits of application of Eq. 1 hinted at in previous studies (for example, Volz 1954). By using a numerical model of the attenuation coefficient of a dispersion (Eq. 4 in Mie theory: Integration of particle size-dependent patterns for a power-law PSD) they showed that Eq. 1 is replaced by the following nonlinear form:
| sD = 3 + γ - 0.5 exp(-6γ) | (2) |
by assuming a finite range of the particle diameter (0.01 to 300 µm), more representative of the attenuation measurements than a semi-infinite range (0 to ∞) used in previous studies. The nonlinearity of the relationship between the slopes of the power-law PSD and the attenuation coefficient spectrum has recently been confirmed by Ackleson (2006), who used a similar Mie-based model for light-absorbing particles with a finite particle diameter range. Given that a particle diameter range representative of light attenuation measurements for specific natural dispersions may be different than that assumed in the model of Boss et al (2001), deviations from Eq. 2 may be expected.
Eq. 2 changes minimally when the absorption coefficient of the particles is non-zero but is spectrally constant, except for the large values of the slope, sD, of the power-law PSD, i.e. for high relative concentrations of the small particles. However, when the particulate absorption spectrum contains strong peaks (typical, for example, of phytoplankton in natural waters) marked deviations of the scattering coefficient spectrum from the power-law due to anomalous dispersion of the refractive index of the particles can be expected. However, a power-law model for the attenuation coefficient is perturbed less in such a case because the effect of anomalous dispersion on scattering can be partially compensated for by the absorption spectrum structure. For this reason, attenuation coefficient spectra are preferable to scattering coefficient spectra when inferring the slope of the power-law PSD. This is convenient since the attenuation of light is easier to measure (at least currently) than scattering. Nevertheless, for the maximum accuracy, one should consider wavelength ranges that do not contain strong absorption peaks when inferring the power-law PSD slope from an attenuation spectrum.
Eq. 1 and Eq. 2 of this topic can, with the caveats discussed above and in the supporting derivation topic be used to determine the slope, sD, of the power-law PSD of a dispersion from the slope of the power-law of the scattering coefficient spectrum of the dispersion and vice versa. By adding Eq. 1 in VSF at the small angles for natural dispersions and power-law slopes to the inversion scheme, one can approximate the wavelength dependence of the scattering coefficient by using measurements of the small-angle VSF and vice-versa.
See also: VSF at the small angles for natural dispersions and power-laws.
| CITATION: Twardowski M., Jonasz M. 2006. Power law in spectral scattering and attenuation by natural dispersions (www.tpdsci.com/Tpc/ScaCfSptPwLwNatDsp.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 07-Sep-2006 Modified: 31-Aug-2006 Peer-reviewed: 13-Nov-2006 |
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