Layers of quasi-horizontally oriented ice crystals in cirrus clouds observed by a two-wavelength polarization lidar

Anatoli Borovoi; Yurii Balin; Grigorii Kokhanenko; Iogannes Penner; Alexander Konoshonkin; Natalia Kustova

doi:10.1364/OE.22.024566

1. Introduction

Optical and microphysical properties of cirrus clouds are of great importance for numerical models of Earth's radiative budget and, consequently, climate [1]. Therefore vertical profiles of their microphysical characteristics, i.e. sizes, shapes and spatial orientation of ice crystals constituting the clouds, are widely studied by ground-based [2–6] and space-borne [7–9] lidars. In lidar measurements, the common backscatter coefficient depends on both the number density of ice crystals and their microphysical parameters. To retrieve only the microphysical parameters, one needs to resort to such dimensionless quantities as the depolarization, color, and lidar ratios, where the lidar ratio is also referred to as extinction-to-backscatter ratio. Here, for example, the depolarization ratio is used to discriminate between water-drop and ice-crystal clouds where small magnitudes of the depolarization ratios are associated with the water-drop clouds. Of course, this statement is true for water clouds only when the depth of penetration is low enough that the depolarization ratio is not increased by multiple scattering. However, such discrimination fails if the ice crystals are quasi-horizontally oriented since both water drops and horizontally oriented crystals give small depolarization ratios. Thus, lidar studies of ice crystal orientation become a challenging problem of the atmospheric optics. We note that light scattering by the horizontally oriented crystals is usually referred to as the specular reflection that is usually observed by vertically pointed lidars.

It is obvious from the physical point of view that if a cirrus cloud is represented by, say, horizontally and randomly oriented ice crystals, these crystals would not be mixed uniformly in space. Indeed, the vertical inhomogeneities inherent to any clouds including their finite vertical thickness are caused by some aerodynamics forces that are obviously dependent on the altitude. These aerodynamics forces impact on both number density and orientation of the crystals. Therefore the horizontally oriented crystals should mainly appear as some horizontal layers. Observations of such layers were described in [5,10]. An excellent review of the aerodynamics forces resulting in the horizontal ice crystal orientation is presented, for example, in [5].

In our recent paper [10], we reported lidar observations of the layers of quasi-horizontally oriented crystals. To identify them, two conjoint criteria were used. Namely, the backscatter signal in such a layer becomes a sharp ridge while the depolarization ratio is turned into a deep minimum. Alternatively, a backscatter signal ridge could also be caused by a sharp increase of the crystal number density. In this paper, we eliminate this ambiguity by simultaneous measurements of both the depolarization ratio at the wavelengths of 532 nm and the color ratio at two wavelengths of 532 nm and 1064 nm, both quantities being not dependent on the crystal number density. In addition, we accompany the experimental observations with our theoretical assessments that are in rather good agreement.

2. Attenuated color ratio and the depolarization ratio

The range-corrected lidar signal can be written as

X (z) = C β (z) \exp [- 2 τ (0, z)],

where z is the altitude, C is the calibration constant,

β

is the backscattering coefficient and

τ

is the optical depth, and where the quantities

β

and

τ

consist of the molecular, aerosols, and cloud components:

β = β_{m} + β_{a} + β_{c}

,

τ = τ_{m} + τ_{a} + τ_{c}

. Above the top

z = H

of a cirrus cloud, we assume that only the molecular backscatter is essential. Therefore for

z > H

we have

X (z) \to x (z) = C β_{m} (z) \exp [- 2 τ_{m} (0, z)] \exp [- 2 τ_{a} (0, H) - 2 τ_{c} (0, H)] .

The molecular parameters

β_{m} (z)

and

τ_{m} (0, z)

are found from the standard atmosphere model. Then a superposition of the experimentally measured function

X (z)

on the function

x (z)

determines the constant

C \exp [- 2 τ_{a} (0, H) - 2 τ_{c} (0, H)] \equiv C^{'}

. This calibration creates the dimensionless backscatter

y (z)

determined as

y (z) = \frac{X (z)}{x (z)} = \frac{β_{m} (z) + β_{a} (z) + β_{c} (z)}{β_{m} (z)} \exp [2 τ_{a} (z, H) + 2 τ_{c} (z, H)],

that corresponds to the altitude backscatter profile in the cloud at

z < H

and it is equal to one at

z > H

. We note that this quantity is also often called the attenuated scattering ratio.

The cirrus cloud backscatter coefficient $β_{c} (z) = n (z) σ (z)$ is the product of the crystal number density $n (z)$ and the average backscattering cross section of one crystal $σ (z)$ . Note that the value $σ (z)$ is an average over both spatial crystal orientations and a statistical ensemble of different crystal sizes and shapes. In this product, only the quantity $σ (z)$ describes the cloud microphysics. Therefore if the cirrus backscatter coefficient $β_{c} (z) = n (z) σ (z)$ is found at two wavelengths as $β_{c}^{(1)}$ and $β_{c}^{(2)}$ , we can remove the crystal number density $n (z)$ by division. The ratio $β_{c}^{(1)} / β_{c}^{(2)}$ is reduced to the conventional color ratio $χ (z) = σ^{(1)} (z) / σ^{(2)} (z)$ reflecting the microphysics profile in cirrus clouds.

In this paper, we use an approximate procedure for a retrieval of the color ratio from lidar signals. Namely, we take a quotient of the functions $y (z)$ of Eq. (3) obtained at two wavelengths

χ^{'} (z) = \frac{y^{(1)} (z)}{y^{(2)} (z)} \frac{β_{m}^{(2)} (z)}{β_{m}^{(1)} (z)} = \frac{β_{m}^{(1)} (z) + β_{a}^{(1)} (z) + β_{c}^{(1)} (z)}{β_{m}^{(2)} (z) + β_{a}^{(2)} (z) + β_{c}^{(2)} (z)} \exp [2 τ_{a}^{(1)} (z, H) - 2 τ_{a}^{(2)} (z, H)] .

Here, in the exponential factor, the optical depth of a cirrus cloud is the integral

τ_{c} (z, H) = \int_{z}^{H} n (z^{'}) σ_{e} (z^{'}) d z^{'}

where

σ_{e}

is the extinction cross section of one crystal. Since crystal sizes essentially exceed the given two wavelengths, the light scattering problem is described by the physical-optics approximation [11]. This approximation gives the simple result

σ_{e}^{(1)} = σ_{e}^{(2)} = 2 s

where s is the averaged area of crystal projection. Consequently, the exponential factor for cirrus clouds in Eq. (4) disappears. The fraction factor in Eq. (4) is approximately reduced to the color ratio

χ (z)

under the conditions

β_{a} < < β_{c}

and

β_{m} < < β_{c}

that are usually true. As a result, the value

χ^{'} (z)

proves to be the attenuated color ratio where the attenuation is caused by only aerosols.

The incident lidar radiation is usually linear or circular polarized. In the polarization measurements, a receiver divides the backscattered light into two components. A component is called parallel if its polarization coincides with the initial polarization both for linear and circular initial polarizations. The orthogonal component is perpendicular. The values $δ (z) = X_{⊥} (z) / X_{| |} (z) = σ_{⊥} (z) / σ_{| |} (z)$ are called the linear $δ_{l}$ and circular $δ_{c}$ depolarization ratios, respectively. These quantities directly extract the cirrus cloud microphysics from lidar signals. In general, any polarization measurements are determined by the Mueller matrix. If the crystals have a plane of symmetry, their orientation distribution is symmetric relative to the vertical, and a lidar points vertically, then the Mueller matrix M becomes diagonal [12]:

M = σ {1; 1 - d; - 1 + d; - 1 + 2 d} .

Here only one quantity d called the polarization parameter determines any polarization measurements. In particular, the conventional linear

δ_{l}

and circular

δ_{c}

depolarization ratios are connected with the polarization parameter d by the simple equations [12,13]:

δ_{l} = d / (2 - d),

δ_{c} = d / (1 - d) .

3. Case studies

We have performed simultaneous measurements of the attenuated color ratio and the polarization parameter in cirrus clouds in Tomsk, Russia, by use of the two-wavelength polarization lidar LOSA-S described earlier [14]. Two cases observed on 30.03.2013 and 14.04.2013 are presented in Figs. 1(a) and 1(b), respectively. Here the red and green curves correspond to the dimensionless backscatter of Eq. (3) for the wavelengths of 1064 nm and 532 nm, respectively. These values are indicated by the lower scale in Fig. 1.

Fig. 1 Altitude profiles in cirrus clouds observed on 30.03.2013 (a) and 14.04.2013 (b): the dimensionless backscatter (red and green) at the wavelengths 1064 nm and 532 nm, respectively (lower scale); the attenuated color ratio (black) and the polarization parameter (blue), (upper scale).

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The upper scale corresponds to both the attenuated color ratios $χ^{'}$ (black curves) calculated by means of Eq. (4) and the polarization parameter d (blue curves) determined by Eq. (5). The polarization parameter was measured for both linear and circular polarization of the incident radiation that was described earlier [10,14]. We note that the Mueller matrix of Eq. (5) for the total lidar signal is a sum of the molecular, aerosols, and cloud Mueller matrices. Though the polarization parameter for molecules and aerosols is rather large, the measured quantity d corresponds to mainly clouds because of relatively small values of the molecular and aerosols backscattering cross-sections.

The cirrus clouds extended in the altitude intervals of 7 km – 9 km (Fig. 1(a)) and 5.5 km – 8 km (Fig. 1(b)). In the first case we can unhesitatingly identify a layer of quasi-horizontally oriented crystals with a thickness of about 500 m within the altitude interval of 7.5 km – 8 km. Such identification follows from three attributes: the dimensionless backscatter for any wavelength reveals a sharp ridge while both the polarization parameter and the attenuated color ratio show deep minima. In the second case, these three attribute identify a thinner layer with the thickness of about 200 m in the altitude interval of 6.5 km – 6.7 km. Though the deep minimum for the color ratio is here shifted a bit from this interval, it can be caused by a poor statistical support. This shift would be clarified in future.

4. Theoretical assessments

Any theoretical assessments for lidar signals returned from cirrus clouds meet certain difficulties. The reason is that the standard numerical methods based on the exact Maxwell equations fail to solve the problem of light scattering by single ice crystal. Here the large ratio of the crystal sizes to the given wavelengths makes great demands to computer capacities. The opposite limit case of geometric optics leads to a singularity at backscattering created by the corner-reflection effect [15] that rejects the geometric-optics approach, too. Consequently, only the physical-optics approximation is effective for this light scattering problem [16]. Such a physical-optics code was recently developed and applied to this problem by the authors [17]. In this section, we compare the experimental data presented in Fig. 1 with our calculations of the cirrus cloud microphysical parameters obtained by our physical-optics code for such common shapes of the ice crystals as the hexagonal ice plates and columns in the case of their arbitrary orientation.

The hexagonal ice plates and columns (Fig. 2) can be characterized by only the diameter D of the hexagon facets while their heights h are found from the empirical equations (see, e.g., [18]). In particular, we consider the plates of $D = 10 μm$ , $100 μm$ , $300 μm$ with the heights of $h = 5.7 μm$ , $16 μm$ , $26.2 μm$ , respectively, and the columns of $D = 10 μm$ , $30 μm$ , $50 μm$ , $100 μm$ with the heights of $h = 10 μm$ , $73.6 μm$ , $198.6 μm$ , $567.8 μm$ . The main axis of the hexagonal prisms is assumed to go through centers of the hexagonal facets and the normal N is taken along the main axis.

Fig. 2 Polarization parameter d for arbitrarily oriented hexagonal ice plates (a) and columns (b). The solid and dashed curves correspond to the wavelength of 532 nm and 1064 nm, respectively, the diameters D of the hexagonal facets for both plates and columns are marked by the color: D = 10 µm (red), D = 30 µm (lilac), D = 50 µm (black), D = 100 µm (green), and D = 300 µm (blue).

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A hexagonal plate is oriented horizontally if $β = 0$ where $β = arc \cos (z \cdot N)$ , z is the zenith direction (|z| = 1), and $0 \leq β \leq π / 2$ . A hexagonal ice column, on the contrary, is assumed to be horizontally oriented if $arc \cos (z \cdot N) = π / 2$ . Therefore we determine the tilt angle as $t = β$ for the plates and $t = π / 2 - arc \cos (z \cdot N)$ for the columns. Then arbitrary orientations of the plates and columns can be determined by the truncated Gaussian probability distribution of the tilt angles

p (t) = \exp (- t^{2} / 2 t_{e f f}^{2}) / \int_{0}^{π / 2} \exp (- t^{2} / 2 t_{e f f}^{2}) \sin β d β .

Here the cases

t_{e f f} = 0

and

t_{e f f} < < π / 2

correspond to horizontally and quasi-horizontally oriented crystals, respectively, while the case

t_{e f f} \geq π / 2

describes the quasi-randomly oriented crystals. In addition to the tilt, a crystal orientation is also determined by the rotation angles

α

and

γ

relative to the zenith and the main axis, respectively. For simplicity, we assume uniform distributions of these angles in the intervals

0 \leq α < 2 π

and

0 \leq γ < 2 π

. As a result, the orientation distribution of the crystals becomes axially symmetric around the vertical z.

The polarization parameters calculated for these arbitrarily oriented crystals and a vertically pointing lidar are presented in Fig. 2. We see that a sharp decrease of the polarization parameter for the quasi-horizontally oriented crystals takes place only for the plates. Consequently, we conclude that the layers of the quasi-horizontally oriented crystals observed in our measurements and presented in Fig. 1 should consist of mainly plate-like crystals. Also we see that the polarization parameters are practically the same for both wavelengths. Therefore any simultaneous measurements of the depolarization parameters for these two wavelengths are not informative if we observe plate-like quasi-horizontally oriented crystals.

The sharp ridge in the dimensionless backscatter within this layer shown in Fig. 1 is explained by alignment of the plate-like crystals near the horizontal plane at these altitudes. Let us estimate this increase of backscatter. The backscattering cross section of a horizontally oriented plate-like crystal of any transversal shape is easily found within the framework of physical optics. Indeed, if a plate with the transversal area S is normally illuminated, the backscattered light leaving the particle surface is just a plane-parallel beam propagating in the backward direction. Assuming an intensity of the incident field of one, then the energy flux of the beam is equal to R·S where R is the reflectance. Strictly speaking, the reflectance R oscillates with plate height h and wavelength λ because of interference; however we can ignore this effect since an averaging over crystal heights taking place in the clouds suppresses these oscillations. During its propagation to a lidar receiver, this plane-parallel beam undergoes the Fraunhofer diffraction being transformed into a spherical wave. Here the energy flux R·S is distributed over the scattering directions with a probability density function corresponding to the Fraunhofer pattern. As known, the probability density in the center of the Fraunhofer pattern is equal to $S / λ^{2}$ [19]. Therefore the backscattering cross section for a strictly horizontal plate $t_{e f f} = 0$ is described by the simple equation

σ_{h o r} = S^{2} R / λ^{2} .

For the opposite case of randomly oriented hexagonal ice plates, we obtained recently by means of numerical calculations the following expression: $σ_{r a n d} \approx 0.007 D h^{2} / λ$ [17]. Substituting both the empirical expression $h \approx 2 \sqrt{D} (in microns)$ [18] and the normal reflectance of an ice plate as $R \approx 0.04$ , we obtain $σ_{h o r} / σ_{r a n d} \approx 0.7 D^{2} / λ (in microns)$ . This explains the large increase of backscatter if the plates are aligned. For example, this ratio is about one thousand for $D = 30 μm$ .

Let us go to the color ratio calculated for these wavelengths $χ = σ (1064 nm) / σ (532 nm)$ . For the horizontally oriented plates $t_{e f f} = 0$ of any transversal shape, Eq. (9) gives the following magnitude of the color ratio $χ = {(532 / 1064)}^{2} = 0.25$ that fits to the numerical data presented in Fig. 3.

Fig. 3 Color ratios for quasi-horizontally oriented hexagonal ice plates (solid curves) and columns (dashed curves). The diameters D are marked by the color as in Fig. 2.

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This color ratio is less than unity because of the energy flux R·S of the backscattered beam is distributed over the scattering directions mainly within the cone of the angle $λ / D$ . Therefore a contribution to a receiver is less if the wavelength is larger. However, if a plate is tilted, the ratio of the contributions becomes changed. Indeed, a tilt t of a crystal plate shifts the diffraction pattern detected by a lidar receiver at the angle $2 t$ . Here a lidar signal obtained for the shorter wavelength can be already negligible while it is still essential for the longer wavelength. As a result, the color ratio should increase. Figure 3 shows the numerical data demonstrating a sharp increase of the color ratio with the effective tilt $t_{e f f}$ for the quasi-horizontally oriented plates. However, an averaging over plate tilts is equivalent to an integration of the Fraunhofer diffraction pattern. If the total Fraunhofer diffraction pattern has been integrated, we come back to the energy flux R·S that is not dependent on the wavelengths. This fact leads to the saturation of the color ratio $χ \to 1$ with increasing effective tilts $t_{e f f}$ according to Fig. 3.

Thus, our calculations support the experimentally obtained data that the color ratio for the plate-like quasi-horizontally oriented ice crystals should reveal a deep minimum like the depolarization ratio. However, a depth of this minimum should be not large as compared with the depolarization ratio. Though the experimental data gives the attenuated color ratio, a difference between the color ratio and attenuated color ratio should be not large since a role of aerosols in Eq. (4) is not essential. In Fig. 3, for comparison, the color ratios for quasi-horizontally oriented hexagonal ice columns are also presented by the dashed lines. We see that though the decrease of the color ratio at $t_{e f f} \to 0$ takes place for the columns too, such a minimum is not much pronounced.

5. Conclusions

Our experimental data obtained by a two-wavelength polarization lidar show that the quasi-horizontally oriented ice crystals in cirrus clouds should appear as some horizontal layers. Three attributes, i.e. a sharp ridge of the backscatter and deep minima of the depolarization and color ratios, seem to be a reliable indicator of such layers. Our theoretical assessments of these quantities confirm the experimental data.

Acknowledgments

The authors thank Dong Liu and Heli Wei for stimulating discussions. This work is supported by the Russian Foundation for Basic Research under Grants no. 12-05-00675a and 13-05-00096a, the RF President grant on the support of leading scientific schools NSH-4714.2014.5, the RF Ministry of Education and Science under the program rising competitiveness of the TSU, and in part by the Russian Science Foundation (Agreement no. 14-27-00022).

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Layers of quasi-horizontally oriented ice crystals in cirrus clouds observed by a two-wavelength polarization lidar

Abstract

1. Introduction

2. Attenuated color ratio and the depolarization ratio

3. Case studies

4. Theoretical assessments

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (9)

Optics Express