Effective Radius

 

  

Figure 5.8: Fraction of light polarized after single-scattering of unpolarized incident radiation as a function of the effective size parameter, $\alpha _{\rm eff}$, and scattering angle.

The effect of varying the value of $r_{\rm eff}$ is now examined. Model calculations were performed for values of 0.1 to 0.3 $\mu$m in 0.05 $\mu$m increments while holding $v_{\rm eff}$ constant at 0.17. Results are shown in Figure 5.7 at 750 nm. The radiances were largely insensitive to $r_{\rm eff}$ with a maximum difference of 5% between $r_{\rm {eff}}=0.15~\mu$m and 0.10 $\mu$m. In general, differences between neighbouring curves were at the 3% level or smaller. The main reason for this is that until the particles are small enough that they begin to behave as Rayleigh scatterers, the phase function varies slowly with size except in the forward and, to a lesser extent, the backward scattering directions. The transition to Rayleigh scattering at 750 nm occurs near $r_{\rm {eff}}=0.10~\mu$m. Comparing the 0.10 $\mu$m and 0.30 $\mu$m radiances, the largest difference is about 30% which occurs at EA $=15^{\circ}$. Differences decrease to approximately 20% near EA=0. Polarization is observed to be much more sensitive. Between 0.1 and 0.15 $\mu$m, there is a maximum difference of 0.05 near ${\rm EA}=0$. For larger aerosols, this difference decreases to less than 0.02. In addition, the shape of the polarization curve is observed to vary. For small aerosols, the characteristic increase near ${\rm EA}=-2^{\circ}$ is absent at 0.10 $\mu$m while it is larger for 0.30 $\mu$m.

As the effects of single-scattering should be evident at 750 nm, it is worthwhile to examine the single-scattering polarization, -P₂₁/P₁₁ (assuming initially unpolarized light). As a function of the effective size parameter, $\alpha_{\rm {eff}}=2\pi r_{\rm {eff}} / \lambda$, and scattering angle, the -P₂₁/P₁₁ ratio is shown in Figure 5.8 assuming initially unpolarized light. For scattering angles between 70-110$^{\circ }$ and $\alpha _{\rm eff}$ between 0.75-3.0, the polarization changes very rapidly. This is the region where Rayleigh scattering is approached. For scattering angles near the forward and backward directions, the variation in polarization is much smaller. In fact, theory states that the polarization state is identically zero (for spherical particles) in the forward and backward directions, irrespective of the size of the scatterer. At 750 nm, the range over which $r_{\rm eff}$ is sensitive is 0.1-0.25 $\mu$m, the same range that was observed in Figure 5.7. It is also evident that beyond 0.35 $\mu$m (or $\alpha_{\rm eff}>3.0$), polarization changed very slowly. Shifting the size range over which the polarization is sensitive towards larger particles, as has been observed following a large volcanic eruption, would necessitate accessing longer wavelengths which are not measured by the CPFM instrument. The steep valley of negative polarization, located at 165$^{\circ }$, is also quite sensitive to $\alpha _{\rm eff}$ between 1.5 and 3.5. This feature is a glory effect, the result of light impinging on the particle at near grazing incidence which sets up surface waves on the particle and acts to focus light in the near-forward and near-backward directions (van de Hulst, 1957). This hill becomes much steeper for smaller values of variance; reducing $v_{\rm eff}$ to 0.07 would lead to a minimum in polarization of approximately 0.65. This feature would be ideal for the retrieval of aerosols between $r_{\rm {eff}}=0.18$ and $0.42~\mu$m. Unfortunately, this geometry requires a solar zenith angle of at least 65$^{\circ }$.

 

Table 5.3: Summary of how polarization varies with a changes in $r_{\rm eff}$ of 0.05 $\mu$m for three geometries.

 

	Geometry
	$\theta_o=30^{\circ}$	$\theta_o=75^{\circ}$	$\theta_o=65^{\circ}$
	$\phi-\phi_o=280^{\circ}$	$\phi-\phi_o=0^{\circ}$	$\phi-\phi_o=180^{\circ}$
Range in $\Theta_{\rm ss}$a	72-98$^{\circ }$	0-30$^{\circ }$	140-170$^{\circ }$
$\Delta LP(0.15,0.10)^b$	0.06	<0.01	0.08
$\Delta LP(0.20,0.15)$	0.04	<0.01	0.06
$\Delta LP(0.25,0.20)$	0.03	<0.01	0.05
$\Delta LP(0.30,0.25)$	0.01	<0.01	0.05

^a The smallest angle in the range corresponds to EA $=15^{\circ}$; the largest corresponds to EA $=-15^{\circ}$
^b $\Delta LP(0.15,0.10)=LP(r_{\rm eff}=0.15~\mu{\rm m}) - LP(r_{\rm eff}=0.10~\mu{\rm m})$

Based on the discussion above, two other geometries have been investigated. The first, $\theta_o=75^{\circ}$ and $\phi-\phi_o=0^{\circ}$, results in scattering angles near the forward direction. The second, $\theta_o=65^{\circ}$ and $\phi-\phi_o=180^{\circ}$, gives scattering angles through the glory, as discussed above. A summary of the polarization results are given in Table 5.3 in terms of the difference in polarization for uplooking angles between consecutive values of $r_{\rm eff}$ studied. For near-forward scattering angles, the difference in polarization was minimal and very close to zero. Scattering angles through the glory resulted in rapidly varying polarization, even for larger aerosol sizes. The difference between these geometries is striking. These single-scattering polarization maps can be quite useful in determining which geometries are best suited to retrievals and, perhaps, may even be used in a rough estimate of the effective variance.