The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles

Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

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Contents lists available at ScienceDirect

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Journal of Quantitative Spectroscopy & Radiative Transfer

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journal homepage: www.elsevier.com/locate/jqsrt

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The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles

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Osku Kemppinen a,b,n, Timo Nousiainen a, Hannakaisa Lindqvist c,d

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a

Earth Observation, Finnish Meteorological Institute, Helsinki, Finland Department of Applied Physics, Aalto University, Espoo, Finland Department of Physics, University of Helsinki, Helsinki, Finland d Colorado State University, Fort Collins, CO, USA b c

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a r t i c l e i n f o

abstract

Article history: Received 7 March 2014 Received in revised form 5 May 2014 Accepted 6 May 2014

The impact of dust particles surface roughness on scattering is investigated using model particles with realistic shapes based on stereogrammetry, up to size parameter 10. The surface roughness is introduced at the particle surfaces by randomly adding or subtracting volume elements in random surface locations with a Monte Carlo method, in such a way that the total number of volume elements comprising the particle is approximately conserved. Assuming an ensemble of randomly oriented particles, roughening seems to decrease the backscattered intensity, reduce the horizontal and vertical polarization, especially in the backscattering hemisphere, and increase diagonal and circular polarization at all scattering angles. The asymmetry parameter and single-scattering albedo are virtually unaffected, while lidar ratio and linear depolarization ratio are increased significantly, especially for large size parameters. These results are qualitatively similar for all particles and the impact of roughness on scattering is clearly related to the amount of roughening. & 2014 Published by Elsevier Ltd.

Keywords: Mineral dust Light scattering Surface roughness

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1. Introduction Mineral dust particles, originating mainly from deserts and other exposed arid regions, are important constituents of the Earth's atmosphere [1]. They can exert a considerable impact on radiation, influencing the radiative balance and remote sensing of the Earth–atmosphere system, e.g., [2,3]. In addition, they act as freezing nuclei and sometimes also as condensation nuclei, contributing to the global water cycle and indirectly to radiation through cloud formation [4,5]. Furthermore, they can be important fertilization

n Corresponding author at: Earth Observation, Finnish Meteorological Institute, P.O. Box 503, 00101 Helsinki, Finland. Tel.. þ358 504 634385. E-mail addresses: [email protected], [email protected] (O. Kemppinen).

agents in iron-limited ecosystems such as surface ocean waters [6] and contribute to air quality and health. The radiative impacts of dust depend not only on their concentrations and spatial distributions, but also on the dust particles' single-scattering properties, which in turn depend on their sizes, shapes and compositions [7,8]. Regarding the shape, it has been realized that the dust particles' small-scale surface roughness may also be quite important for their single-scattering properties [9–20]. In general, it appears that the introduction of surface roughness tends to smoothen the angular dependence of the scattered intensity, promote positive linear polarization at side-scattering angles, and alter the size dependence of scattering. However, different studies often differ in the details, suggesting a complex interplay between the roughness elements and the host particle. Clearly, the

http://dx.doi.org/10.1016/j.jqsrt.2014.05.024 0022-4073/& 2014 Published by Elsevier Ltd.

61 Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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impact of surface roughness on scattering depends on the refractive index, host particle shape, and the particle size relative to the wavelength of radiation [7]. When modelling the single-scattering properties of mineral dust, a shape model for the dust particles needs to be established. Historically, these have often been based on simple mathematically defined geometries (e.g., spheres, spheroids, ellipsoids, Chebyschev shapes), or more complex, phenomenological shape models that attempt to mimic the appearance of real dust particles [21]. Recently, stereogrammetry of electron microscope images was adapted in an attempt to derive the real shapes of a small number of real dust particles [22]. As surface roughness is recognized to be a radiatively important parameter in dust particle modelling, and the impact of roughness appears to depend on the model particles' overall shapes, it appears prudent to investigate the impact of surface roughness on scattering using realistically shaped dust particles. To this end, we will use the dust particle shapes derived by Lindqvist et al. [22] and apply a roughening scheme to manipulate their surface roughness characteristics.

23 2. Theoretical aspects 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

In the investigation of the surface roughness on scattering, we will focus on the elements of the scattering matrix S, which relate the Stokes vectors for the incident and the scattered wave in a scattering event such that 2 30 1 0 1 Ii S11 S12 S13 S14 Is 6 7B C B Qs C Q S S S S 6 7 B C 22 23 24 1 6 21 B C 7B i C; ð1Þ B C¼ 7 B C U S S S S @ U s A k2 d2 6 32 33 34 5@ i A 4 31 Vi S41 S42 S43 S44 Vs where subscripts ‘i’ and ‘s’ refer to incident and scattered waves, respectively; Stokes parameter I describes the intensity, Q and U the linear polarization, and V the circular polarization of the wave; k ¼ 2π=λ is the wavenumber related to the wavelength λ, and d the distance from the scatterer. The scattering matrix thus contains all the information about the scattering event that can be carried by the scattered wave. Assuming that the particles are randomly oriented, and particles are mirror symmetric or particles and their mirror particles are present in equal numbers, the scattering matrix simplifies into six independent elements [23]: 2 3 S11 S12 0 0 6 7 0 0 7 6 S12 S22 7 S¼6 ð2Þ 6 0 S34 7 0 S33 4 5 0 0  S34 S44 Indeed, even when all these conditions are not strictly valid, the scattering matrices seem to closely conform to this form for ensembles of complex particles, such as dust [21]. Therefore, for this application, it is sufficient to consider only the effects on these non-zero matrix elements. The diagonal elements govern how the I, Q, U and V are preserved in the scattering event. The off-diagonal elements govern how the pairs I and Q, and U and V, transform into each other.

We also investigate the effect of roughening on several scalar quantities. The asymmetry parameter g describes how the scattered intensity is divided between forward (θ o 901) and backward (θ 4 901) hemispheres, where θ is the scattering angle, that is, the angle between the propagation directions of the incident and the scattered radiation. Asymmetry parameter value of unity would indicate that all radiation is scattered into the exact forward direction, whereas g of 1 would mean that all radiation is scattered into the exact backward direction. The definition of g is as follows: Z π 2π g¼ 2 sin θ cos θS11 ðθÞ dθ; ð3Þ k C sca 0

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where C sca is the scattering cross-section, a measure of the scattering power of the particle. Single-scattering albedo, ω, describes the proportion of scattering to total extinction in a single scattering event. Extinction is the sum of absorption and scatterings' crosssections and is related to total incident energy reduction by the particle. Value of unity for ω means that all of the radiative energy is scattered and none is absorbed, whereas a zero value means that all of the radiation is absorbed and none is scattered. Calculating ω is straightforward from the scattering and total extinction crosssections:

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C sca : C ext

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When lidars are used to retrieve aerosol properties, a quantity called lidar ratio, R, is required. The lidar ratio is the ratio of the extinction to backscattering cross-section, and can be calculated as

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R¼

C ext k C ext ; ¼ C back S11 ð1801Þ

ð5Þ

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where S11 ð1801Þ refers to the value of S11 at θ ¼ 1801. Linear depolarization ratio, δL , is another quantity of interest in aerosol lidar retrievals due to the sensitivity of S22 to the shape of the particle. For example, spherical, isotropic particles have S11 ¼ S22 ) δL ¼ 0. Linear depolarization ratio is defined as

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δL ¼

S11 ð1801Þ  S22 ð1801Þ ; S11 ð1801Þ þ S22 ð1801Þ

ð6Þ

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with S22 ð1801Þ referring to the value of S22 at the exact backscattering direction, θ ¼ 1801.

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3. Modelling approach

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3.1. Particle shapes

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The shapes used as the basis of the surface roughness analysis are stereogrammetrically retrieved shapes of four single mineral dust particles described by Lindqvist et al. [22]. The particles were selected from the samples collected during the SAMUM campaign over Morocco [24]. They were imaged with the scanning-electron microscope from two different directions, and based on these images, the topographies of the particle surfaces were retrieved using stereogrammetry. Since the undersides of the particles were

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Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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not imaged, the lower hemispheres were constructed by assuming the lower hemisphere mirroring the upper hemisphere. If necessary, the mirrored hemisphere was scaled with respect to the vertical axis, to conserve the aspect ratio of the particle, which could be measured. The surface presentation was then converted into a volumetric one to be used in light-scattering simulations based on the discrete-dipole approximation, where scattering is computed by integration of the electric fields induced by small, discrete volume elements called dipoles. The lightscattering method used is described in more detail in Section 3.3. The elemental compositions of these particles were obtained using energy-dispersive spectroscopy, and further, their mineralogical composition was deduced and implemented on the retrieved model shape volume elements with different refractive indices. In the present study, the surface structure of the particles is stochastically modified, rendering the refractive indices on the surface ambiguous. Therefore, for simplicity, here we assume the particles to be homogeneous, composed of an effective refractive index; as Lindqvist et al. [22] showed, this simplification is of minor significance for these particles. While the stereogrammetric particles present shapes much more realistic than what is commonly used in light scattering studies for dust, it is nevertheless prudent to point out the inherent limitations of the current implementation of the stereogrammetric approach. Firstly, the method has a varying resolution of the surface details, deteriorating towards the edges where only the coarsest surface structures are retrieved successfully. Secondly, the construction of the other hemisphere by mirroring can cause unnaturally smooth, cut surfaces, which may become more realistic with the introduction of roughened surfaces. The particles used are described in more detail in Lindqvist et al. [22]. Here we adopt their naming convention for them: the particles are called Cal I, Dol I, Agg I, and Sil I. In addition, for comparison, we consider an artificial cube-shaped reference particle, henceforth called Cube. The refractive indices of the particles are as follows: Cal I: 1.619 þ 0.00124i, Dol I: 1.620 þ 0.000000467i, Agg I: 1.615 þ 0.00355i, Sil I: 1.589 þ 0.00278i, and Cube: 1.6 þ 0.001i. The value y ¼ jmjkd, where m is the refractive index, k is the wavenumber and d is the dipole size, is used to evaluate the applicability of the discrete-dipole approximation. The largest y value for the particles in this study was roughly 0.56, which is well below the commonly cited DDA accuracy limit of yr 1 [25]. Furthermore, Lindqvist et al. [22] tested different dipole resolutions for these exact particles (excluding the cube) and concluded that the single-accuracy lattices used also in this study are satisfactory regarding the accuracy of the simulations.

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Because the stereogrammetric method is unable to resolve the smallest details in the particles, the surfaces of the model particles derived may be unnaturally smooth. Consequently, the model particles may fail to reproduce

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the single-scattering properties of the actual particles accurately. To study the possible effects of the potentially missing small-scale surface roughness on scattering, we have developed a method to roughen the particle surfaces. The method is applied to discrete-dipole representations of particles, and can, in principle, treat any particle in a DDA format. The method is based on colliding the target with vectors, henceforth called rays. The operation of the method can be divided into three parts. First, the starting points of the rays as well as their directions are determined. Second, the particle surface elements they collide with, if any, are located. Third, the particle surface is modified in the collision point. A more detailed description of each of the three steps is given below. Each particle is contained in a regular cubic lattice. Initially, the lattice dimensions are equal to the particle maximum dimensions, but to allow for the collisions to grow the particle, we pad the original lattice with empty elements. The padding chosen for this study was 16 elements in each dimension, eight on each side. Empirical tests with the parameters used in this study showed this to be more than sufficient to contain all the new dipoles formed by collisions. This padded particle lattice is contained within an infinite space lattice, which allows us to use a lattice traversing method even outside the padded particle lattice. In the first step, we begin by creating a bounding surface around the particle from which the rays will be spawned. In this study, we use a sphere as the bounding surface. The sphere is positioned in such a way that the center is in the geometric center of the particle. The radius of the sphere is equal to half of the diagonal length of the padded lattice. A larger sphere would allow for a more homogeneous roughening for very non-spherical particles, but with an additional computational cost due to slower lattice traversing and a smaller probability for the rays to hit the target. The radius described above was determined to be sufficient by visual inspection of the roughened particles. It should be noted that the bounding surface was used only for spawning the collision rays and has no direct effect on the particle geometry or the scattering itself. The second part of the first step is to sample random point pairs from the bounding surface. These pairs form the start and the end points of the rays we use to collide with the particle, and thus also determine the direction of rays. In the second step, for each of the rays, we find which surface element of the DDA particle it collides with, if any. This is done by traversing the space lattice from the start point in the direction of the ray, and checking if elements in the path are occupied, i.e., not empty. The first occupied element encountered is the collision element. Each collision has a 50% chance to either add or to remove dipoles. This is because we want the total dipole amount of the particle to remain essentially unchanged. Because the ray directions are random and not specifically directed at the particle surface in any way, some portion of the rays will miss. Missed rays will have no effect. We sample new rays from the sphere surface until a

Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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pre-defined number of collisions is reached. In this work, we consider three values for the number of required collisions: 1500, 3000 and 6000. The cases are independent, that is, the first 1500 collisions of the 3000 collision case were not the same collisions as those in the 1500 collision case. In the third step, regardless of whether the effect was to add or to remove dipoles, we find out the volume of the effect in dipoles, i.e., how many dipoles will be changed. The volume is determined by taking a random sample from a normal distribution. In this study the mean volume of the effect, nmean , was 8 dipoles, pffiffiffi and the standard deviation of the volume, nSD , was 8  2:83 dipoles. This distribution was chosen to have a clear effect in the scattering matrix, while not overly changing the overall shape or producing anomalous structures. In the general case the effect volume should be somehow proportional to the total size or the surface area of the particle. Since in this study all the particles were approximately the same size and shape, we decided to use the same effect volume distribution for all of them. If the sample from the volume distribution is n, then we take n (rounded to the nearest integer) closest dipoles to the collision dipole that are empty (when adding dipoles) or occupied (when removing dipoles). Then, the affected dipoles are converted to either occupied or empty. Essentially, this causes either a mound or a crater on the surface. Repeating this process of colliding the target with random rays causes the particle to accumulate these small mounds and craters, which will eventually form a roughened surface. With nmean ¼ 8, most of the affected dipoles are within a few unit lengths of the surface, so the overall shape of the particle does not change significantly. After a sufficient number of collisions, the particles are ran through a filter to remove any isolated clusters, i.e., groups of (or single) dipoles that are not attached to the main particle. This is done simply by starting from one dipole, here selected as the geometric center of the particle, and traversing recursively all the orthogonal neighbors. After there are no more unvisited neighbors, the dipoles that were not visited are removed. We have observed that the number of dipoles does not change significantly in the process of roughening the particles and then removing the isolated clusters. Table 1 shows the total number of dipoles and the relative change in the number of dipoles for each roughening stage for each particle. We see that in most cases the number of dipoles is changed by less than a percent, the 6000 collision case of Cal I being the only exception. This is important, because the scattering model scales the particle size according to the number of dipoles it has. Therefore, changing the number of dipoles significantly would change the effective size of the particle compared to the unroughened case, which would make it difficult to interpret which effect caused the change in scattering. We also note that, in case the dipole amounts were changed significantly, it would be possible to compensate for it by modifying the probabilities for mounds and craters. The illustrations of the particles produced in this way are shown below. Fig. 1 includes the four stereogrammetric particles as well as the Cube. Each row illustrates

Table 1 Effects of roughening on the volumes and the affected surfaces of the particles. The number of dipoles, the percentage change in the number of dipoles compared to the unroughened case, and the surface area affected by roughening are shown. Particle # of collisions

# of dipoles

Cal Cal Cal Cal

I I I I

0 1500 3000 6000

100,563 0 100,559  0.00 100,736 þ0.17 99,450  1.11

0 51.11 74.58 91.01

Dol Dol Dol Dol

I I I I

0 1500 3000 6000

101,322 0 101,565 þ0.24 100,846  0.47 101,926 þ0.60

0 57.75 76.89 92.56

Agg Agg Agg Agg

I I I I

0 1500 3000 6000

100,710 0 100,841 þ0.13 100,565  0.14 100,205  0.50

0 51.72 73.54 89.86

0 1500 3000 6000

100,779 0 100,685  0.09 100,457  0.32 100,605  0.17

0 38.18 58.62 80.49

0 1500 3000 6000

103,823 0 103,879 þ0.05 103,527  0.29 104,031 þ0.20

0 49.74 73.11 90.12

Sil Sil Sil Sil

I I I I

Cube Cube Cube Cube

Percentage volume change

Affected area percentage

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the effects of the roughening for one particle. The leftmost particle is the unroughened case, and the roughening is progressing from left to right. We can also calculate how much of the surface of the particle was affected. This gives us an estimate of the amount of roughening. For each surface dipole, we say that it was affected if either it itself or any of its initially empty neighbors in the six Cartesian directions were changed at any point during the roughening process. That is, a surface dipole with only one empty neighbor (smooth surface case) will be affected if either the surface dipole itself is empty in the end of the simulation, or the one empty neighbor is occupied when the simulation is finished. This is equivalent of the surface dipole either being erased by a crater, or being hidden by a mound. In this case, only the end state matters, because the main point of interest is how much the shape differs in the end, specifically how much the surface has changed. Table 1 shows the percentage of affected surface area for each roughening stage for each particle, where being affected is defined as above. We see that the general progress is such that 1500 collisions produce approximately a 50% surface change, 3000 collisions produce approximately a 75% surface change and 6000 collisions produce approximately a 90% surface change. That is, doubling the number of collisions roughly halves the amount of non-affected surface dipoles. Sil I is a clear outlier, with each stage having consistently less affected surface as the other particles. This is possibly due to larger local curvature that dominates much of the surface. In high curvature areas the affected surface area is smaller than in flat areas. As another possible factor, since the shape of the surface from which the collision rays were determined was

Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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Fig. 1. Visualizations of all the roughness stages of all the particles used in this study. Each row includes all the roughness stages of one particle, progressing from the unroughened particle on the far left to the most roughened case on the far right.

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123 Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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a sphere, some dipoles of Sil I have a larger probability of being collided with compared to other particles. This is due to the relatively large solid angle of the dipoles closest to the bounding sphere, as well as their shadowing effect. This kind of an effect would cause some dipoles to experience frequent collisions, while the rest of the dipoles would collide with less. Regardless, even Sil I follows approximately the same progression: N=M

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Aaff is the affected percentage, l is a constant individual to each particle, N is the number of collisions and M is another constant. With M as 1500 and l ¼0.62 for Sil I and l¼0.5 for other particles, we get 38%, 62% and 76% for Sil I, and 50%, 75% and 88% for other particles as the predicted affected surface areas. While this simple identity is not very accurate, it is useful for roughly estimating the affected surface area values for other N.

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3.3. Scattering model

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For the scattering calculations we used the program ADDA, version 1.2 MPI [26]. ADDA implements the discrete-dipole approximation, which allows the user to simulate light scattering by an arbitrarily shaped collection of dipoles. The method has the great advantage of providing complete freedom regarding the shape of the target, which is obviously a requirement for a study such as this. The accuracy is limited by both the determination of the real shape of the particle and by the quantization resolution of that shape into discrete dipoles. The former issue is discussed in Section 3.1. The latter issue is related to the computing resources, since the number of dipoles used has a large effect on the computation time. In this study each of the particles were quantized into roughly one hundred thousand dipoles. The scattering of each particle was averaged over 8192 random orientations. ADDA was run on the Finnish Meteorological Institute supercomputer Voima, using 16 computer cores per simulation. Additionally, 10 concurrent simulations were run in parallel to reduce the total run time. With this setup, the total amount of CPU time used was approximately 6000 h.

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4. Results

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The scattering quantities will be analyzed, on one hand, as a function of the size parameter x ¼ 2πr=λ, where r is the particle radius, in this case the radius of a volumeequivalent sphere. On the other hand, we consider quantities integrated over a pre-defined size distribution for which we use here the lognormal size distribution with the geometric mean radius r g ¼ 0:4 μm and the geometric standard deviation sg ¼ 2:0. This distribution is adapted from [22] with the following special requirements:

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 Provide a realistically looking size distribution where

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small particles dominate the number concentration. Provide such weights for the size parameters x A ½0:5; 10 included in the simulations that they all contribute.

 Provide as high an effective radius as possible given the

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modest size coverage, so that the results would have relevance for atmospheric applications.

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The effective radius provided by the distribution, r eff ¼ 0:82 μm, is somewhat below the supermicron values typically encountered in the atmosphere, but should provide us with reasonable estimates for the sensitivity of scattering on roughness, and on the persistence of the effects over size integration. Simulating the impact of roughness on scattering at larger particle size parameters would require much more computing power or more advanced computational methods, and are left for the future. As an additional issue, larger particles require a finer discretization resolution, i.e., more dipoles. In addition to further increasing the computational power requirements, it would also require additional testing regarding whether the effects of roughening stay constant regardless of the discretization resolution. In this study we are mainly interested in how the roughening changes the values of the elements of the scattering matrix. This is studied in two ways: First, we take a look at the differences between the scattering matrix elements of the unroughened particles and the roughened particles as a function of the size parameter. Second, we investigate the impact of roughening on the scattering matrix elements integrated over the size distribution. In addition to studying the effects of roughening on the realistically shaped particles, we also studied a cubeshaped artificial particle with 473 ¼ 103,823 dipoles. One reason for using the cube was to see how various stages affect a particle that initially had completely smooth surfaces. Using an artificial particle also made it easier to set the roughening parameters such as to generate a clearly roughened surface but not to disturb the original overall shape too much.

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4.1. Size dependence of the effect due to roughness

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It is interesting to know how the impact of roughness on scattering depends on the size parameter. In Fig. 2 we have calculated the integral over the scattering angle θ of the absolute difference between the unroughened particle and the different stages of roughening. This gives us a quantity for each scattering matrix element, total scattering difference, to describe the total effect of roughening on scattering. We show the total scattering difference as a function of the size parameter for all three roughening stages. On one hand, the total scattering differences for S11 =C sca are shown. S11 =C sca is used instead of just the S11 to see the relative effect instead of the absolute one. Otherwise, the increasing differences would be mostly due to the increased scale of the S11 values, and different size parameters would be not comparable. On the other hand, we show H, which is a sum of the total scattering differences over the other five matrix elements divided by

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Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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Fig. 2. The absolute differences of θ-integrated scattering matrix elements caused by roughening. S11/Csca and a heuristic sum H of the other element ratios to S11 are shown. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

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the corresponding S11, called the heuristic sum. That is,

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H¼∑

Z x

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π

    Sx ðθÞ S0;x ðθÞ    S ðθÞ  S ðθÞdθ ; 11 0;11

ð8Þ

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where x A f12; 22; 33; 34; 44g indicates the scattering matrix element, S refers to the roughened scattering matrix in question and S0 refers to the unroughened scattering matrix. The heuristic sum is, in some sense, the total effect of the roughening on all of the polarization elements of the scattering matrix. The effect on S11 =C sca rises fairly exponentially, i.e., linearly in the logarithmic scale of the plot, for the whole size parameter range. For Agg I and Cube, the effect on S11 seems to become almost constant for large sizes. For the other particles, the size parameter 10 was not enough to stabilize the effect. The phenomenon is similar for the heuristic sum. The absolute effect increases as a function of the size parameter for all the stages of roughening. In addition, the differences between the different stages of roughening also increase in the absolute sense. Here we recall that the heuristic sum is the sum of the absolute values of the differences of all the polarization elements of the scattering matrix. The fact that the heuristic sum looks consistent with the increasing size parameter as well as the increasing roughness stage indicates that the behaviors of total scattering differences for each individual scattering matrix element are qualitatively similar: roughening affects the scattering matrix elements more for larger size parameters. Indeed, although not shown here, the only exception to this is  S12 =S11 , for which the largest effect was seen at size parameter 2. Even for  S12 =S11 , the increase in roughening systematically increases the total scattering difference.

4.2. SD-integrated individual scattering matrix elements

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Here we show the scattering matrix elements as a function of the scattering angle θ. We show the value of S11 and ratios  S12 =S11 and S22 =S11 . We also analyzed S33 =S11 ,  S34 =S11 and S44 =S11 , but the effects for them are just briefly outlined, not shown. Each figure contains the values for the original particle as well as three different stages of roughness, as described in the Section 3.2. The figures show the size-distribution integrated values for each independent scattering-matrix element. Since most of the scattering elements seem to behave similarly for each of the particle shapes, we discuss the results element by element rather than particle by particle, noting exceptional particles when needed (Fig. 3). For S11, the most prominent effects are seen in the backscattering hemisphere and in particular close to the exact backscattering direction, where S11 decreases with increasing roughness. The high sensitivity of the backscattering direction on roughness has been previously noted by, e.g., Kahnert et al. [16]. In the forward hemisphere the impacts are much weaker and also opposite in sign: roughness slightly increases the scattered intensity. Cal I and Agg I seem to be the least affected particles. Looking at Fig. 1, they seem to be the particles with the most curved areas and the least sharp edges; thus most rough to begin with. The impact of roughness for the Cube is qualitatively similar to those for stereogrammetric particles. Compared to S11,  S12 =S11 shows a great deal of more variability, both in the elements themselves and in the effect of roughening on them. Noting the different scales of the figures of different particles, we see that roughening has the largest absolute effect on Dol I, Sil I and Cube, while Cal I and Agg I are affected less. The effects are

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Fig. 3. S11 values for each of the particles shown as a function of the scattering angle. Each of the different roughness cases, including the original unroughened case, are shown separately. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

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Fig. 4.  S12 /S11 values for each of the particles shown as a function of the scattering angle. Each of the different roughness cases, including the original unroughened case, are shown separately. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

similarly split between curved and edged particles as was with S11 above. Except for Sil I, which already shows a strongly positive  S12 =S11 , roughness tends to make  S12 =S11 more positive. Again, the impacts for the Cube are similar to those of the stereogrammetric particles (Fig. 4). The impact of roughness on S22/S11 seems similar to that for S11, but with a somewhat more pronounced effects. There is a systematic but relatively small positive

effect from the forward-scattering direction until θ¼901, peaking at around 601. The exception is Sil I, for which this effect is not small but very significant. The effect is again reversed for θ 4901 for which the increased roughening decreases the value of the element. The largest negative effect is seen at 1801 for all the particles except Agg I and Sil I for which the effect is a bit larger near 1401. Overall, Dol I and Cube are again the most affected particles, followed by Cal I.

Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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When considering the effects the scattering matrix elements have on the polarization of the scattered light, we note that only the upper left quadrant values of the scattering matrix, Sx ; x A f11; 12; 22g, have an effect on the unpolarized and horizontally/vertically polarized light, as represented by the Stokes vector components I and Q. Correspondingly, only the lower right quadrant values of the scattering matrix, Sx ; x A f33; 34; 44g, can impact the diagonally or circularly polarized components of the scattered light, that is, the Stokes vector components U and V. Therefore, given that all of Sx ; x A f11; 12; 22g are decreased by roughening at most of the scattering angles, roughening seems to decrease the overall intensity of the scattered unpolarized and horizontally/vertically polarized light. The more smoothly curved particles Cal I and Agg I had initially smaller values for the upper left quadrant scattering matrix elements, and for those particles those matrix elements were also decreased less than for the other particles. In other words, the reduction of the scattering matrix elements that affect the Stokes vector elements I and Q is largest for the particles that initially had large values for the corresponding scattering matrix elements (Fig. 5). The impact of roughness on the elements S33/S11, S34 =S11 and S44/S11 is the following. S33/S11 is increased by roughening for all scattering angles and all particles. All the particles show a monotonous increase in effect from the forward-scattering direction until 901. Backwards from that the effect starts to decrease slightly, however staying significant for the whole backscattering hemisphere. The roughening effect on  S34 =S11 is generally to decrease it, the largest effect being seen around 901. The relative effect can be large near θ ¼ 901, up to 100% increase to the unroughened value. S44/S11 is similar to S33/S11, the roughening effect being always positive as well. The effect is

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close to zero in the forward-scattering directions, and largest at directions near the scattering angles 1351 and 1801. Therefore, as noted above, while the roughening effect on the scattering matrix elements affecting I and Q of the Stokes vector was negative, it is positive for the elements affecting U and V.

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4.3. Scalar scattering quantities The results on the asymmetry parameter of the particles are shown in Fig. 6. We show the values for the original particle as well as the three stages of roughening as a function of the size parameter. The asymmetry parameter itself increases rapidly from the smallest size parameter to approximately 3–4, where it stabilizes. For Dol I, Cube and, to a smaller extent, Cal I there is a significant dip around the x ¼ 6  2π, where the particle radius is the same as the wavelength of the incident radiation. Roughening seems to have only a very small but consistent effect on the asymmetry parameter. For all the particles and all size parameters above 2–3 roughening increases the asymmetry parameter. The magnitude of the effect seems consistently related to the number of collisions for each particle and size. Compared to the asymmetry parameter, the range of the single-scattering albedo as the function of the size parameter is much smaller, allowing a strongly truncated y-axis to be used in the plots. Still, the impact of roughness is barely visible in the plots, making the impact extremely weak. This is consistent with the findings in [16,18] that the increased absorptivity increases the impact of roughness on ω. Of course, if the particles are non-absorbing, ω  1 and there cannot be any effect. Thus, our findings should not be assumed valid for more strongly absorbing

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Fig. 5. S22/S11 values for each of the particles shown as a function of the scattering angle. Each of the different roughness cases, including the original unroughened case, are shown separately. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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Fig. 6. The impact of roughening on the asymmetry parameter g for the model particles considered as a function of size parameter and the amount of roughening. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

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Fig. 7. The impact of roughening on the single-scattering albedo ω for the model particles considered as a function of size parameter and the amount of roughening. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

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dust particles. The effect is nevertheless again consistent in the sense that increasing the number of collisions increases the effect for all particles and sizes. More roughening seems to increase the albedo. For large sizes of Agg I the 6000 collision case has roughly the same effect as one unit change in the size parameter (Fig. 7).

Regarding the lidar quantities, roughening has a substantial effect on the lidar ratio that clearly exceeds that for the asymmetry parameter or the single-scattering albedo. For the small size parameters up to x¼2, the effect is virtually zero; for larger x, the effect is roughly constant. The effect seems to be always positive, roughening leading

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to higher values, increasing with more collisions. For large size parameters, roughening can have a very significant relative effect due to a small absolute value of the lidar ratio for the unroughened case. For example, the lidar ratio for size parameter 10 Sil I approximately doubles with 6000 collisions. The Cube also experiences a large effect for large size parameters, but the effect at large size parameters for other particles is only of the order of 10%. The lidar ratio changes are mainly due to changes in S11 ð1801Þ (see Eq. (5)). C ext values are affected only slightly in the roughening process (not shown), a few percent at most, and much less for x o 8. However, it is important to note that the sign of the effect also supports the increase of the lidar ratio, and seems to increase with larger x. Therefore, for x much larger than studied here, the changes in C ext might also contribute to the increase in the lidar ratio significantly. As was the case with the lidar ratio, the linear depolarization ratio is fairly sensitive to roughening. The effects are seen mainly for size parameters larger than 4, where roughening leads to higher values and larger effects with increasing x. The percentage changes in δL can be very significant, especially for large sizes (Fig. 8). For the Cube and Sil I particles, δL can increase by more than 100% due to roughening at large x. Dol I and Cal I also show roughly 50% increase, while for Agg I the increase is only 10–15%. In general, the more the δL is affected, the larger the size parameter. The strength of the increase for each particle type is clearly correlated with δL value for the original particle: the larger the value, the smaller the increase. Thus, particle to particle differences are smaller for roughened particles. Except for Sil I with a very small starting value, other particles typically have δL  0:5 after roughening.

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5. Summary and conclusions

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In this work we modified surfaces of four realistically shaped dust particles to investigate the impact of surface roughness on the dust-particle optical properties. For comparison, we also considered an artificial cube-shaped particle. The roughening was performed by bombarding the particles with virtual rays that created small mounds or craters upon impact. The bombardment parameters were chosen such that the overall shape was not disturbed significantly, and the volume of the particles remained approximately within 1% of the original. Three different cases for roughening were tested: 1500, 3000 and 6000 collisions. The optical properties for each roughening stage of each particle, including the original unroughened case, were simulated using a discrete-dipole approximation. The scattering simulations were calculated for 20 size parameters: 0.5, 1, 1.5,…,10. The results obtained were analyzed in two ways. First, we looked at the differences in scattering matrix elements separately for each size. Second, we considered the sizeintegrated values for the scattering matrix elements (Fig. 9). The method chosen for roughening the particle surfaces seems to perform as intended. As seen from Table 1, roughening does not change the volume of the particles appreciably. In addition, as seen in Fig. 1, the particles retain their overall original shapes fairly well. Therefore, the changes seen in the various scattering quantities can be attributed to arising solely from changing surface roughness characteristics. The impact of roughness on scattering seems to increase with the progressing stage of roughening. Although only S11 and the heuristic sum of the other

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Fig. 8. The impact of roughening on the lidar ratio R for the model particles considered as a function of size parameter and the amount of roughening. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

Please cite this article as: Kemppinen O, et al. The impact of surface roughness on scattering by realistically shaped wavelength-scale dust particles. J Quant Spectrosc Radiat Transfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.05.024i

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Fig. 9. The impact of roughening on the linear depolarization ratio δL for the model particles considered as a function of size parameter and the amount of roughening. (a) Cal I, (b) Dol I, (c) Agg I, (d) Sil I, and (e) Cube.

elements were shown in this work, each individual element has the same consistent behavior. In addition, scattering seems to have a larger effect for large than small size parameters. Due to the modest size parameter coverage in this work, it cannot be established whether there is such a size parameter where the scattering effect would start to decrease. Due to this high degree of regularity, the effects of the roughening persist even after integrating over a size distribution. For these size-integrated quantities, shown in Section 4.2, roughening seems to decrease S11, S12/S11 and S22/S11, and increase S33/S11, S34/S11 and S44/S11. Therefore, roughening seems to dampen the unpolarized and horizontally/vertically polarized scattering, and enhance diagonally and circularly polarized scattering. Overall, the diagonal scattering matrix elements are affected the most in the backscattering direction, and the off-diagonal elements are affected the most at sizescattering angles. In most cases roughening either increases or decreases the value of the element similarly for all the scattering angles. Asymmetry parameter and single-scattering albedo were affected only slightly by roughening. The effect, however small, was that of increasing for both of these variables, and was larger for larger size parameters. In addition, the effect systematically strengthened with increasing roughness. The lidar ratio and the linear depolarization ratio experienced a significantly larger effect from roughening. They were both increased by roughening, similar to the asymmetry parameter and the single-scattering albedo. Specifically for large size parameters, the relative effect on both of these variables can be very significant: from tens to a hundred percent. It is very noteworthy that the effect of roughening is qualitatively similar for all of the scattering variables for all of

the particles, including the reference cube. This shows that the surface roughness can be important for scattering both for regular and irregular shapes. The impact, however, depends on the corresponding values for the unroughened shapes and can, in principle, be negative or positive. It is therefore possible that the roughness has no effect for some quantities for some particles. Previous literacy summarizes the impact of roughness on scattering as a decrease in S11, an increase in  S12 =S11 at the side-scattering angles, and an increase in S22 =S11 especially at backscattering angles [27]. Our results regarding the first two conclusions are similar, but the effect on S22 =S11 seems to be of the opposite sign. We speculate that this difference in dependence is either due to sensitivity of S22 =S11 to details of the roughening method or because of the different S22 =S11 values in the initial unroughened case. In this work we have developed a method for roughening surfaces of dust particles, and shown that surface roughness, or lack thereof, has a significant impact on light scattering, even for particles with complex shapes. The roughening method is versatile and applicable for a wide range of DDA particles. However, since the method works only for DDA particles, it also means that the size parameter coverage is fairly modest: light simulations for size parameters larger than approximately 20–30 would take a very long time even with the help of supercomputers. Based on our results, the impact due to roughening would likely be larger at size parameters larger than considered here.

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Acknowledgments

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This research has been funded, in part, by the Academy of Finland (Grant 255718) and the Finnish Funding Agency for Technology and Innovation (Tekes; Grant 3155/31/

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2009). Maxim Yurkin is acknowledged for making his ADDA code publicly available.

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