1. Introduction

Cirrus clouds are important components of the atmosphere which essentially modulate the energy budget of the Earth-atmosphere system. At present, cirrus clouds are sources of considerable uncertainties in predicting the climate change. Therefore, a lot of efforts have been undertaken for studying both the optical and microphysical properties of cirrus clouds (a review and references can be found, e.g., in [1]). The microphysics of cirrus is size, shape and spatial orientation of the ice crystals constituting the clouds, these quantities being greatly variable both spatially and temporally in the real atmosphere.

Among the other devices, lidars are perspective instruments used for studying cirrus clouds from both the ground and space. In these studies, the cirrus microphysics is mainly inferred from the elastic backscattering coefficient measured at one or several of the conventional wavelengths of 0.355, 0532, and 1.064 µm. The majority of the lidars are also the polarization ones measuring the depolarization ratio or, rarely, 4 × 4 scattering or the Mueller matrix [2] at a fixed wavelength. Besides, the Raman lidars [3,4] and the high spectral resolution lidars (HSRL) [5] can be explored for detecting the extinction coefficient in cirrus clouds as well. Recently, the unique measurements of the ice water content (IWC) in cirrus using the Raman lidars have been reported in [6]. The backscatter-to-IWC ratio was also used for the joint interpretation of the lidar and radar signals [7].

In spite of the fact that there are a lot of such measurements, a retrieval of the cirrus microphysics from lidar signals still remains a challenging problem for the lidar community. The reason for this is that the problem of the elastic backscattering by the ice crystals has not been satisfactorily solved until now. Here the main obstacle in solving this problem is that the typical size of the ice crystals of about 30µm - 100µm is much larger than the conventional lidar wavelengths. In this case, the standard numerical methods solving the problem of light scattering by nonspherical particles [8] become computationally expensive. Among the approximate methods applicable to this problem, the improved geometrical-optics method (IGOM) is widely used [1]. However, the IGOM algorithm for the randomly oriented hexagonal ice crystals had underestimated the backscatter phase function as compared with the experimental data and an additional backscattering peak was suggested in [9,10] to avoid the inconsistency. One of the difficulties at numerical calculations of the backscatter is that several millions of crystal orientations are needed in the case of the random orientation as it was recently emphasized in [11,12].

At present, the physical-optics approximation [12,13] proves to be a fast and reliable method for calculating the backscatter for the ice crystals of cirrus clouds that takes correctly into account both the diffraction and interference phenomena at light scattering. Using this method, the backscattering Mueller matrix for a number of typical crystal shapes (hexagonal columns and plates, bullets, and droxtals) has been recently calculated in the interval of crystal size from 10 µm up to 1000 µm at arbitrary crystal orientations for three conventional lidar wavelengths. These data are available in [14]. Note that the smooth regular crystals presented in [14] create the backscattering peak because of the corner reflection [13] while it was recently shown that the backscatter proved to be strongly sensitive to both the geometrical irregularities and surface roughness of the crystals [9,10,15,16]. The backscatter for such irregular crystals is a subject for further calculations.

In this paper we show that the backscattering Mueller matrixes as the functions of the crystal size in the case of randomly oriented particles can be presented with good accuracy by the power laws. Thus, the simple analytical functions presented below can be used instead of the data bank [14]. These analytical functions essentially simplify any averaging over the crystal size values in the statistical ensembles of the crystals in the cirrus clouds. As an example, four backscatter ratios widely used in lidar research have been presented in this paper as the functions of the modal size for the above crystal shapes.

2. Mueller matrix and the backscatter ratios

The light scattered by any particle is fully described by the Mueller matrix. If ice crystals are randomly oriented or an ensemble of preferably oriented crystals reveals its rotational symmetry around the lidar line-of-sight, the backscattering Mueller matrix has view

Calibrated lidars can measure the following values

Dividing the quantities of Eq. (2) by each other we obtain the backscatter ratios characterizing only the microphysics of the ice

3. Mueller matrix versus crystal size

The shape of an ice crystal is determined by a lot of geometric sizes describing the crystal faces. According to the experimental data [18,19], the geometric sizes of the ice crystals can be linked by some power laws presented in the second column of Table 1. Consequently, in the atmospheric studies, the crystal size is usually characterized by only one value. This value is either the radius of the sphere of the equivalent volume or the maximum dimension inside the crystal denoted later as l. For example, if the crystals are hexagonal columns or plates with the diameters D of the hexagons and the length L between the hexagonal faces, the maximum dimension is equal to $l={(D^2+L^2)}^{1/2}$.

Table 1. The power laws for the backscattering Mueller matrixes

View Table

Figure 1 shows the logarithm of the backscattering matrix elements versus the logarithm of crystal size in the interval from 10 µm up to 1000 µm for the conventional lidar wavelengths of 0.355, 0.532, and 1.064 µm. Here the dots and crosses are the values M₁₁ and M₂₂, respectively, taken from our data bank [14] that has been calculated by means of our physical-optics code [20].

 

Fig. 1 The Mueller matrix versus crystal size (the angles ψ, θ₁, and θ₂ shown in Figs. 1c and 1d were equal to 28°, 32.35°, and 71.81°, respectively). The refractive index is 1.3249, 1.3116, and 1.3004 for the wavelengths of 0.355, 0.532, and 1.064 µm, respectively.

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In Fig. 1, one can see that the calculated data are well interpolated by the straight lines. It proves that the backscattering matrixes for the randomly oriented crystals can be described with good accuracy by the power functions of the crystal size

Table 1 presents the coefficients of Eq. (4) and Fig. 2 shows the relative difference $\epsilon =(y-y_0)/y_0\cdot 100\%$ where y is the calculated value taken from the data bank [14] and $y_0$ corresponds to Eq. (4) for the non-zero elements M₁₁ and M₂₂ of the Mueller matrix.

 

Fig. 2 Relative difference between the calculated Mueller matrix and the power laws (in percent). The dots and crosses correspond to the elements M₁₁ and M₂₂, respectively, and the color corresponds to the legend of Fig. 1a.

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The power laws of Eq. (4) are indirectly caused by the power laws that appear approximately for the geometrical cross section of the crystals under the empirical relations between the geometrical sizes of different crystal faces presented in the second column of Table 1. Of course, the geometrical cross section and the backscattering cross section are not proportional. This fact is mostly manifested in the case of the hexagonal plates in Fig. 1b where the geometrical cross section should be obviously a monotonically increasing value while the backscattering cross section reveals the weak oscillations. Nevertheless, the power laws prove to be good approximations since the oscillating deviations shown in Fig. 2 will be partly compensated at averaging over the crystal size. By the way, we notice that the oscillations for the plates are caused by the variable contributions from the ray trajectories with multiple reflections from the base faces shown in the lower part of Fig. 1b.

It is important to notice that the spectral dependence of the backscatter shown in Figs. 1a-1c is associated with the corner reflection effect described earlier [21]. However, this dependence disappears in Fig. 1d for the droxtals of large size. It means that the contribution of the corner-reflection terms decreases for more complicated crystal shapes.

4. Backscatter ratios versus the modal crystal size

In the real atmosphere, sizes of the ice crystals are distributed within a wide interval where some effective size characterizes the microphysics. To investigate the dependence of the backscatter ratios of Eq. (3) on the effective size, we suggest that the crystals are distributed over the maximum dimensions according to the gamma-distribution

 

Fig. 3 The backscatter ratios of Eq. (3) versus the modal size. The color corresponds to the legend of Fig. 1a.

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In literature, the experimental data for these ratios are few and fragment while the theoretical data are absent at all. According to the data of Fig. 3 obtained for the first time, we see that the depolarization and color ratios prove to be weakly sensitive to the crystal size in the cirrus. On the contrary, the lidar and backscatter-to-IWC ratios could be useful for inferring the crystal size in the clouds.

5. Conclusion

The backscattering Mueller matrix of the ice crystals of cirrus clouds is the basic quantity needed for the lidar community sounding the clouds. On the base of our numerical data obtained by use of the physical-optics-approximation code, this matrix has been presented in the paper as the power functions of the crystal size for some typical crystal shapes. Four commonly used backscatter ratios have been calculated for the first time as the functions of the crystal modal size that can be useful for retrieving the cloud microphysics.