1. Introduction

Marine aerosol is emitted into the atmosphere by wind tearing off wave crests and air bubbles bursting in the oceans. Due to the vast source regions, marine aerosol is one of the most abundant aerosol types in the atmosphere [1]. Marine aerosol influences the climate directly by scattering radiation [2–4] and indirectly by acting as cloud condensation nucleii, which in turn affects cloud reflectivity and precipitation [1]. It also provides large surface areas for heterogeneous chemical reactions [2]. In coastal areas sea salt aerosol plays an important role in corrosion processes of metals and reinforced concrete structures [5]. Sea water contains a mixtures of different dissolved salts, of which sodium chloride is most abundant [6]. The composition of sea salt aerosol particles is determined by the minerals contained in the ocean’s surface layer [3,7–11]. Marine aerosol may further contain biogenic material [12].

With increasing relative humidity sea salt aerosol particles grow by water condensing onto the crystal [13,14]. The condensed water increasingly dissolves the salt crystal. At a relative humidity of approximately $70-74\%$ the main delinquescence point of sea salt is reached [15] and the sea salt crystal becomes fully dissolved in a liquid droplet. Partial dissolution of sea salt occurs already at lower values of RH caused by salts other than NaCl [10,16]. With decreasing relative humidity such a fully dissolved sea salt crystal remains in liquid state until the relative humidity is below approximately $45-50\%$ [10,16]. At this point, the efflorescence point, the sea salt recrystallizes. As a consequence of this hysteresis effect both crystalline salt particles with a water coating and fully dissolved, hence liquid salt aerosol particles may exist between values of relative humidity of $\sim 50$ and $\sim 70\%$. For comparison, pure sodium chloride’s deliquescence point is at $RH=74.0\pm 0.2\%$ and its efflorescence point is at $43.0\pm 1.0\%$ [17]. Thus, between values of the relative humidity between 45 and 70% both spherical droplets containing fully dissolved salt and solid, partially nonspherical salt crystals with a watery hull can exist. From laboratory measurements [18], we know that dry, crystalline salt particles ($RH=10-20\%$) change the polarization state of incident laser light (linear depolarization ratios of $\delta _l=0.21\pm 0.02$ for sodium chloride and $\delta _l=0.08\pm 0.01$ for sea salt), while the polarization state is largely unchanged by dissolved salt at $RH>90\%$ ($\delta _l=0.01\pm 0.001$ for both sodium chloride and sea salt). This change in linear depolarization ratio can be attributed to the transition from cube-like salt crystals to spherical droplets of fully dissolved salt.

A previous study found that the optical properties of dried sea salt aerosol particles including associated uncertainties can be modelled using convex polyhedra, a randomized class of model particles with a cube being the limiting case [19]. As a first objective, our study plans to extend this model by considering sea salt particles with a water coating. To this end, we introduce a model for water-coated salt particles that uses a pseudo-potential model in order to account for the effect that water vapor will predominantly condense onto regions on the salt crystal’s surface with a large number of neighboring atoms, while salt ions in the solid phase and in contact with liquid coating will predominantly go into solution if the ions have few neighbors. The geometries constructed with this model display a gradual transition from film-coated irregular cuboid crystals to more rounded salt cores with a thick spherical liquid coating. We use this rather complex model as a reference geometry, which serves three purposes. First, we average optical properties over several realizations of the randomized geometries, which is closely analogous to what one would observe in nature. Second, by investigating the range in which the optical properties vary among different stochastic realizations of the randomized geometries, we can obtain an estimate of the model’s error variance, which is an important input parameter to inverse methods, such as data assimilation (e.g. [20]). Third, by comparing this rather detailed and morphologically realistic particle to simpler geometries, we can investigate which morphological features make the dominant contributions to the optical characteristics.

In addition to convex polyhedra homogeneous sea salt aerosol particles have previously been modeled using spheres [21–25], cubes [18,23–27], elongated and flattened cuboids [24], superellipsoids [19,28], and Gaussian random cubes [19]. Spherical models usually represent dissolved sea salt, while the other shapes represent dried sea salt. A subset of these studies investigated the linear depolarization ratio [18,19,25–28]. Inhomogeneous, i.e. water-coated sea salt, was modeled with inhomogeneous cube-like and octahedra-like superellipsoids and superellipsoids embedded in a spherical shell [28]. Originally used to calculate single scattering properties, such as the linear depolarization ratios and scattering coefficients [28], these inhomogeneous models were also employed to calculate the direct radiative forcing of inhomogeneous sea salt [29] and polarized radiance at the top of the atmosphere [30].

Off-center inhomogeneities in spherical model particles can give rise to large values of the linear backscattering depolarization ratio, as studies for soot containing aerosol showed [31,32]. Here, using a non-concentric spherical shell model (see Sec. 2.2) the effect of off-center inhomogeneities on the optical properties is investigated.

Our morphologically complex and realistic reference model, which takes random deviations from a cubical shape and the changes due to water condensing onto a salt crystal into account, is computationally demanding. Operational use of lidar observations, such as processing global remote sensing data from satellites, would require a computationally less expensive model. The second aim of this study is to find a simplified model, which can simultaneously simulate optical cross sections and linear depolarization ratio.

A microphysical property hypothesized to have a key impact on the optical properties is the overall (non-)sphericity of the particles. Here both homogeneous and inhomogeneous particles are considered. In order to test this hypothesis model particles, which allow a gradual change between cubical and spherical geometries, are investigated. One particle class used, are homogeneous superellipsoids, another are inhomogeneous superellipsoids (see Sec. 2.3). Superellipsoids were previously found to be a suitable geometry type to model sea salt optical properties [19,28]. As an additional model geometry a cube-sphere hybrid model is introduced. By contrast to superellipsoids, this model results in less pronounced rounding of the cubes’ edges.

To summarise, we address the following questions.

The particle models employed are discussed in Sec. 2. Section 3 discusses the single scattering codes used. The results are presented in Sec. 4 and discussed in Sec. 5.

2. Particle models

The basic idea is to investigate (i) how modeled optical properties of marine aerosol particles change as more and more water is taken up by the particles; and (ii) how modeled optical properties compare under different assumptions of the morphology of dry and water-coated marine aerosol particles. For each model particle the dry-particle size and the salt-volume fraction in the salt-brine composite particle are prescribed. Note that we do not make any assumptions here under which thermodynamic conditions such volume fractions would be encountered in nature. Thus the optics models are completely general and could be applied in conjunction with any parametrizations that provide the amount of water taken up by the particle as a function of temperature and relative humidity.

Four different models are used in this study: convex polyhedra with a pseudo-potential coating model, a non-concentric spherical shell model, a superellipsoidal model, and a cube-sphere hybrid model. Two different types of superellipsoids are used - homogeneous superellipsoids and inhomogeneous superellipsoids.

2.1 Reference model: Convex polyhedra with pseudo-potential coating

This class of model particles will serve as our reference geometry. The main morphological features of both the salt core and the water coating are intended to mimic those of realistic water-coated sea salt particles.

Dried sea salt particles display deviations from a perfectly cubical shape, as can be seen in images shown in [3,12,14,25,33–36]. In a previous study of dried sea salt aerosol particles convex polyhedra were found to be a suitable geometry type for modeling the optical properties of sea salt aerosols and capturing the effects of these shape deviations. Stochastic variations in the model resulted in bias-free, random errors [19] of optical properties, which can be useful for estimating model uncertainties.

Convex polyhedra are created by randomly placing a number of points $N_c$ in a Cartesian coordinate system, and by applying a convex hull around these points [19,37]. The resulting shapes are irregular and their surfaces are composed of plane faces. $N_c$ controls the shape. For a decreasing number of $N_c$ the shapes become increasingly non-cubical up to a point, when they introduce a bias in the optical properties. By contrast, for high values of $N_c$ the variation in the optical properties is small, as the shapes differ only little from a cube. Intermediate values of $N_c$ proved to be the most useful model particles. Following the results in [19] $N_c$ is assumed to be 50, 100, and 250, respectively, and five different stochastic realizations of the geometry per value of $N_c$ are generated. On average convex polyhedra with $N_c=50$ fill out $58.4\%$ of a cube’s volume with an edge equally long as the respective polyhedron’s largest extend along the Cartesian axes. Due to the different stochastic realizations the range of this fraction is between 54.3% and 62.1%. For $N_c=100$ and $N_c=250$ the polyhedra fill the corresponding cubes on average to 66.3% (full range: 64.1-69.2%) and 81.7% (full range: 79.6-84.3%), respectively. This quantifies, how the convex polyhedra become more cube-like and how the shape variation for different stochastic realisations decreases with increasing $N_c$.

Dry or partially coated sea salt particles in moist air can adsorb water to its surface. The water will partially or completely dissolve the salt. Microscopically, water uptake results from the Coulomb interaction of water vapor molecules with molecules or ions on the particle’s surface. This interaction is likely to be stronger at those surface points that have a large number of neighboring molecules/ions. We mimic this effect by use of a simple pseudo-potential model, which has previously been used to mimic surface-tension effects in sulfate-coated soot particles [38]. This model lends itself easily to be used in conjunction with the discrete dipole approximation (see Sect. 3.1). More specifically, we assume that the volume of the particle has been discretized into $N$ equal-size cubic volume cells of side length $d$. We define for each volume cell $i$ on the particle surface a potential

There are several complications in this construction process. The most obvious one is that the addition of liquid water changes the surface of the particle. Thus, the surface potential needs to be re-evaluated periodically. Following the methodology in [38], this is done in a two-step process, which we abbreviate by add_coating:

In this approach we largely follow the suggestions in [38]. However, with the settings given in [38] for $\alpha _1$ and $\alpha _2$, our large water-coated salt particles do not become spherical. Our settings of $\alpha _1=0.004$ and $\alpha _2=0.001$ ensure that the water-coating makes a smooth transition from a nonspherical thin film to a thick spherical coating as more and more water is applied to the surface.

The two-step process that we labeled add_coating would need to be repeated until the total amount of liquid coating material has been applied. However, there is a second complication that we need to consider first. The salt core is water soluble, by contrast to the insoluble soot particles considered in [38]. We assume that the salt solution is saturated, where the solubility of sodium chloride at 283 K has been used, which is 36 g NaCl per 100 ml water. We further assumed that sea salt has a density equal to that of NaCl, 2240 kg/m$^3$, and that the brine coating has a density of 1210 kg/m$^3$. This determines, at each step of the coating process, the number of volume cells that are removed from the salt core, and the number of volume cells that are to be added to the coating due to the addition of the solute. Microphysically, those salt ions on the solid surface in contact with water that experience the weakest Coulomb interaction with their surrounding are most likely to go into solution. We model this by use of the surface-potential model given in Eqs. (1) and (2), where $N_s$ now denotes the number of salt cells that lie on the water-covered part of the solid-particle surface, and where the sum over $j$ extends over all grid cells that contain solid salt. We define an algorithm analogous to the one discussed above, but based on the points of highest potential (corresponding to the weakest Coulomb interaction). We denote this process by dissolve_salt:

Here we set $\beta _1=0.0016$ and $\beta _2=0.0004$.

The complete algorithm starts from a prescribed dry particle size and a target salt-volume fraction for the brine-coated particle. This predetermines the number $N_w$ of water-cells that are to be added (as well as the number of salt-cells that need to be dissolved). Then the iteration proceeds by the following scheme.

DO WHILE $N_w(\mathrm {now}) < N_w$

 add_coating

 dissolve_salt

END DO

$N_w(\mathrm {now})$ denotes the number of water-cells that are presently applied at any given step of the iteration. The important point is that the surface potential, which determines the attachment of water and the dissolution of salt, is frequently updated in this iteration. This is controlled by $\alpha _1$, $\alpha _2$, $\beta _1$, $\beta _2$.

The model geometries are illustrated in Fig. 1 for a polyhedron with $N_c=100$. The values of $f_{\mathrm {vol}}$ decrease from left to right with $f_{\mathrm {vol}}=1.0$ (left), $f_{\mathrm {vol}}=0.8$, $f_{\mathrm {vol}}=0.55$, and $f_{\mathrm {vol}}=0.3$ (right). The original dry particle (left) is generally irregular but similar to a cube, depending on the choice of $N_c$. Adsorption of a small amount of water results in a thin film coating that closely follows the shape of the salt core (second from left). We see already at this early stage of the coating process that the sharp edges of the salt particle become slightly rounded off, because the solid volume cells along the edges, which have few neighbors, have a high surface potential and go quickly into solution. As more water is applied the rounding of the edges becomes more pronounced and the coating approaches spherical shape (second from right). For thickly coated particles both the coating and the salt core become spherical (right). Application of even more water will result in spherical particles in which the salt has become completely dissolved (not shown). This model has a number of free parameters. The settings we chose for our reference model are meant to qualitatively mimic the behavior in real atmospheric sea-salt particles. In particular, if $\alpha _1$, $\alpha _2$ are too large, then the coating does not reach spherical shape.

 

Fig. 1. Example for a coated convex polyhedron ($N_c=100$) with different values of $f_{\mathrm {vol}}$, from left to right - $f_{\mathrm {vol}}=1.0$, $f_{\mathrm {vol}}=0.8$, $f_{\mathrm {vol}}=0.55$, and $f_{\mathrm {vol}}=0.3$.

Download Full Size | PDF

2.2 Non-concentric spherical shell model

Calculations of the energy potential of solid inclusions in liquid droplets suggest, that such inclusions are predominantly moved towards the droplet’s surface [39]. Thus, the inclusions have an off-center position. The impact of the resulting off-center inhomogeneities on the optical properties is investigated using a non-concentric spherical core-shell model, based on [40]. The non-concentric spherical shell models aim at investigating a potential refinement of the coated polyhedra model. The inner sphere represents the salt core, which is encapsulated by a shell representing saturated salt solution. The radii of both spheres correspond to the volume equivalent radii of the entire particle and its solid core as determined with the coated convex polyhedra model. The inner sphere’s center is shifted with respect to the shell’s center, but constrained to always lie within the spherical shell. The relative shift of the core’s center $s_{\mathrm {rel}}$ is expressed as fraction of the shell’s radius. As $f_{\mathrm {vol}}$ decreases, the possible range of $s_{\mathrm {rel}}$ increases.

A similar model was used in [31] for modelling the linear depolarization ratios of coated soot. The non-concentric spherical core shell model used in this study differs in the variable size of the core, reflecting the slowly dissolving salt crystal, from the version used in [31].

A schematic of the model is shown in Fig. 2. With decreasing $f_{\mathrm {vol}}$ (as indicated by the rows, from top to bottom), increasing relative shifts are possible (indicated by the columns, increasing from left to right.)

 

Fig. 2. Schematic of the non-concentric spherical shell model. The salt core is shown in grey and the watery shell in blue. The different rows represent different salt fractions $f_{vol}$ and the columns different relative displacements of the salt core with respect to the watery shell.

Download Full Size | PDF

For the non-concentric spherical shell model values of $f_{\mathrm {vol}}$ ranging from 1.0 to 0.15, with a linear spacing of $\Delta f_{\mathrm {vol}}=0.05$ are considered. In addition $f_{\mathrm {vol}}=0.139$ is also considered. For smaller values, the salt core becomes fully dissolved. The shift of the core $s_{\mathrm {rel}}$ is, starting from $s_{\mathrm {rel}}=0.0$, increased by $\Delta s_{\mathrm {rel}}=0.025$. This results in 228 different combinations of $f_{\mathrm {vol}}$ and $s_{\mathrm {rel}}$, for which the core is placed within the hull.

2.3 Superellipsoids

Superellipsoids have previously been used to model mineral dust [41,42] and sea salt aerosol [19,28,29]. Superellipsoids describe three-dimensional shapes as the product of two superquadratic curves [43,44]. They can be considered a generalization of ellipsoids and are represented in the Cartesian coordinate system by [44]

Here $a$, $b$, and $c$ determine the particle’s extent along the Cartesian axes $x$, $y$, and $z$, respectively. The particle’s shape is specified by $n$ and $e$, the roundness parameters in the north-south (or polar) and east-west (or azimuthal) directions, respectively [41,44]. This makes superellipsoids a highly versatile type of geometry, allowing to obtain cubes, spheres, cylinders and octahedra by just changing the roundness parameters. An explicit parameterization of superellipsoids in spherical coordinates has recently been derived and applied in [19].

In the present study $a=b=c$ is assumed, i.e. the respective aspect ratios are $a/c=b/c=1$. Nevertheless, the term superellipsoids will be used throughout this study. Following the approach by [41] $n=e$ is assumed. The range of $n=e$ is further limited to 0 (cubes) – 1 (spheres). During the process of water adsorption and the ensuing dissolution of the sea salt crystal the shape changes from nearly cubical to spherical (with inclusions) and eventually to a homogeneous sphere. This highlights the potential suitability of superellipsoids to approximate the reference model for different values of $f_{\mathrm {vol}}$ and hence the different resulting overall shapes. Here, two different types of superellipsoids were used - homogeneous and inhomogeneous superellipsoids. Investigating both homogeneous and inhomogeneous superellipsoids allows us to study simultaneously non-sphericity and inhomogeneity and the respective impact on the optical properties. For homogeneous superellipsoids an effective refractive index for a mixture of sodium chloride and a staturated sodium chloride solution is determined using the Maxwell-Garnet mixing rule [45]. Inhomogeneous superellipsoids consist of an outer layer representing a saturated sodium chloride solution and a sodium chloride core. Both the outer layer and the core possess the same roundness parameters. The layer’s thickness is determined by the salt volume fraction.

Figure 3 shows examples for the two types of superellipsoids employed in this study. The roundness parameter increases from left to right, so that the shapes become increasingly spherical. The top row shows homogeneous superellipsoids, i.e., they have a core fraction of $f_{\mathrm {vol}}=1.0$. The bottom row shows inhomogeneous superellipsoids with a core fraction of $f_{\mathrm {vol}}=0.55$. The solid fraction is represented by dark grey and the liquid fraction by light-blue. For both inhomogeneous and homogeneous superellipsoids the calculations are performed for eleven different values of $n$ ranging from $n=0.0$ to $n=1.0$ with $\Delta n=0.1$, and for nine different values of $f_{\mathrm {vol}}$, ranging from $f_{\mathrm {vol}}=1.0$ to $f_{\mathrm {vol}}=0.3$, with $\Delta f_{\mathrm {vol}}=0.1$ as well as for $f_{\mathrm {vol}}=0.55$.

 

Fig. 3. Examples for superellipsoids with two different solid fractions $f_{vol}=1.0$ (top row) and $f_{vol}=0.55$ (bottom row) as used in this study. The roundness parameter $n$ of the superellipsoids increases from left to right.

Download Full Size | PDF

2.4 Cube-sphere hybrids

The superellipsoids are an extremely versatile class of model particles. However, while easy to implement in the discrete dipole approximation, using them in Waterman’s T-matrix method [46] poses some challenges. In that method one employs the surface parameterization $r_{\mathrm {s}}(\theta, \phi )$ of the particle surface, as well as its partial derivatives $\partial r_{\mathrm {s}}/\partial \theta$ and $\partial r_{\mathrm {s}}/\partial \phi$. Here, $(\theta, \phi )$ denote spherical coordinates. This is important, as Waterman’s method is based on the use of vector spherical wave functions defined in spherical coordinates. Neither the implicit parameterization given in Eq. (3), nor explicit parameterizations in other coordinate systems, such as the one found in [44], can be readily used in Waterman’s method. To the best of our knowledge, the first derivation of $r_{\mathrm {s}}(\theta,\phi )$, $\partial r_{\mathrm {s}}/\partial \theta$, and $\partial r_{\mathrm {s}}/\partial \phi$ for superellipsoids, as well as its application in the T-matrix method, has been reported only very recently [19].

A much simpler alternative are cube-sphere hybrids. Let us denote the surface parameterization of a cube by $r_{\mathrm {c}}(\theta,\phi )$, and let $r_0$ denote the radius of a volume-equivalent sphere. We define the surface parameterization of the cube-sphere hybrid by

 

Fig. 4. Cube-sphere hybrid particles, in which the parameter $\eta$ is, from left to right, 0.1, 0.3, 0.5, 0.7, and 0.9. The cube ($\eta =0.0$) and the sphere ($\eta =1.0$) are not shown.

Download Full Size | PDF

As a side-note, we could equally well have defined the hybrid by a linear superposition of the radii, i.e., $r_{\mathrm {csh}}(\theta,\phi ) = (1-\eta ) r_{\mathrm {c}}(\theta,\phi ) + \eta r_0$. As long as both the cube and the sphere have the same volume, the hybrid particle will also have the same volume. However, if $r_{\mathrm {c}}^3\neq r_0^3$, i.e., when the cube and the sphere have different volumes, then the volume of the composite particle and its volume-equivalent radius will be easiest to compute from Eq. (4).

3. Computational methods

To obtain the single scattering properties three different codes are used - the discrete dipole approximation code ADDA and the T-matrix codes MSTM and Tsym.

The single scattering codes return the scattering matrix $F$ and the optical cross sections. From these quantities the linear depolarization ratio in the backscattering direction can be calculated by [47]:

$F_{11}$ and $F_{22}$ denote the (11) and (22) element of the scattering matrix, respectively. The matrix elements are evaluated in the backscattering direction ($\theta =180^{\circ }$).

The extinction-to-backscatter ratio, in context of lidar remote sensing also referred to as lidar ratio, can be calculated by [48]

Here $C_{\mathrm {ext}}$ denotes the extinction cross section. The backscattering cross section is expressed by $\left.\frac {1}{4\pi }C_{sca}F_{11}\right |_{\theta =180^{\circ }}$.

The calculations were performed for a wavelength of $\lambda =0.532\,\mu$m, the second harmonic of Nd:YAG lasers. Nd:YAG lasers are commonly used in lidar remote sensing [49,50]. The refractive index of salt was assumed to be equal to the refractive index of sodium chloride, $m_{NaCl}=1.5484+i0$ [51]. In accordance with the reference model, the liquid hull was assumed to consist of a saturated solution of NaCl in water (brine), for which we used a refractive index $m_{NaCl,sol}=1.3174+i6.19\cdot 10^{-7}$ [52].

3.1 ADDA

The discret dipole approximation (DDA) code ADDA, version 1.3b4 [53,54], was used to calculate optical properties of the coated polyhedra and the inhomogeneous superellipsoids.

The DDA divides the scatterer into a number of small, fully polarisable volume-elements much smaller than the wavelength. These volume-elements are referred to as dipoles. Owing to this dividing the scatterers into dipoles the DDA can treat arbitrarily shaped and inhomogeneous particles. The dipole spacing $d$ was chosen, following uncertainty estimates discussed in [19], so that $|m|kd=0.38$. Here $m$ denotes the complex refractive index and $k$ the wavenumber of incident light. This corresponds to a values of dipoles per wavelength of $dpl=25.5$.

Orientation averaging is required as the scatterers are assumed to be totally randomly orientated. Orientation averaging is performed in ADDA over the three Euler angles $\alpha$, $\beta$, and $\gamma$ using Romberg integration. The set of integration points is defined by the minimum and maximum value and the number of subdivisions $J_{\mathrm {min}}$ and $J_{\mathrm {max}}$. For the coated convex polyhedra $J_{\mathrm {max}}$ for $\alpha$ and $\gamma$ was chosen to be $J_{\mathrm {max}}=4$ and for $\beta$ to be $J_{\mathrm {max}}=3$. With the number of arguments of the integration limited to $2^{J_{\mathrm {min}}}+1$, this results in 2601 discrete orientations. In case of the inhomogeneous superellipsoids a higher angular resolution was used, with $J_{\mathrm {max}}=6$ for $\alpha$ and $J_{\mathrm {max}}=5$ for $\beta$ and $\gamma$. This choice was motivated by the sensitivity study in [19] to obtain a better agreement of the linear depolarization ratios of homogeneous superellipsoids, which was calculated using the T-matrix code Tsym , and inhomogeneous superellipsoids in the shared limiting case of $f_{\mathrm {vol}}=1.0$.

The scattering matrix elements were calculated with an angular resolution of $0.5^{\circ }$. As an iterative solver the quasi minimal residual solver QMR2 was used. Other than that ADDA default settings were used.

3.2 MSTM

The multiple sphere T-matrix code MSTM, version 3.0 [55], was employed to compute optical properties of non-concentric spherical shell particles. This code is based on the superposition T-matrix method. The method and consequently the MSTM code itself can be applied to scatterers consisting of multiple spheres with non-overlapping surfaces. This implies, that the surfaces of any two spheres may at most share a single point. The code iteratively solves the interaction equation and computes the orientation-averaged optical properties from the T-matrix by using analytical solution to the integration over Euler angles.

3.3 Tsym

Optical properties of homogeneous superellipsoids and of cube-sphere hybrid particles where computed by use of the T-matrix code Tsym, version 6.6. This code is based on Waterman’s null-field method (also known as the extended boundary condition method). The code exploits symmetries of the particle geometry by (i) using the commutation relations of the T-matrix to reduce CPU-time requirements [56,57], and (ii) making use of irreducible representations of the symmetry group [58] in order to reduce numerical ill-conditioning problems, but also to further cut down the computation time. Orientation-averaged optical properties are computed by use of analytic expressions as described in [59,60]. The latest release version 5.2 of the code is described in [61] and published in [62]. The cube-sphere hybrid particles have been added in version 5.7, but results obtained with that model are published here for the first time. Superellipsoids have been added in version 6.6. This required us to derive the surface parameterization of superellipsoids in spherical coordinates, as well as its partial derivatives. The derivation as well as a first test of Tsym in conjunction with superellipsoids have been published in [19].

4. Results

4.1 Coated polyhedra

The results obtained with the reference model are shown in Fig. 5. The columns refer to the different optical properties, from left to right: $\delta _l$, $C_{\mathrm {bak}}$, $C_{\mathrm {ext}}$, and $S_p$. The rows refer to the different radii of the dry salt particle $r_0$, with $r_0=0.33\,\mu$m, $r_0=0.67\,\mu$m, and $r_0=1.0\,\mu$m from top to bottom. The colors indicate the arithmetic means over the different values of $N_c$ with $N_c=50$ (orange), $N_c=100$ (green), and $N_c=250$ (magenta). The corresponding error bars indicate the maximum range of results obtained for five stochastic realizations of the geometry for each instance of $N_c$. The arithmetic mean over all values of $N_c$ and all stochastic realizations is shown in blue. Note, that the corresponding blue error bar denotes the standard deviation over all realizations and all values of $N_c$.

 

Fig. 5. Results of the reference model, i.e. depolarization ratio $\delta _l$, backscattering cross section $C_{\mathrm {bak}}$, extinction cross section $C_{\mathrm {ext}}$, and extinction-to-backscatter ratio $S_p$ (columns from left to right) for $r_0=0.33\,\mu$m (top row), $r_0=0.67\,\mu$m (middle row), and $r_0=1.0\,\mu$m (bottom row) at $\lambda =0.532\,\mu$m. The results for $N_c=50$ are shown in orange, for $N_c=100$ in green, $N_c=250$ in magenta. Blue denotes the mean over all three values of $N_c$. Error bars denote the total range and in case of the mean over $N_c$ (blue) the standard deviation.

Download Full Size | PDF

For each value of $r_0$ and $f_{\mathrm {vol}}$ the mean linear depolarization ratio decreases with increasing $N_c$. For $r_0=0.33\,\mu$m and $r_0=0.67\,\mu$m the range of $\delta _l$ also decreases with increasing $N_c$. Both effects can be attributed to the fact that model particles become, on average, more cube-like with increasing $N_c$, as reported in [19]. Regular cubes depolarize less than distorted, irregular cubes. Further, for increasing values of $N_c$ the stochastic deviations from cubic shape become less pronounced than for low values of $N_c$, resulting in a reduced variability in depolarization ratios.

For larger particles, especially for $r_0=1.0\,\mu$m, we observe an initial increase in the linear depolarization ratio as $f_{\mathrm {vol}}$ is reduced from 1.0 to 0.8. This is most likely caused by an increase in particle size without a significant change in the overall shape. This observation is consistent with results reported in [28], according to which a water coating without shape changes increases the linear depolarization ratio for single particles with size parameters of up to $x=15$, which corresponds to $r_{\mathrm {eq}}=1.27\,\mu$m for a wavelength of $\lambda =0.532\,\mu$m. With further condensation on the particle the linear depolarization ratio decreases as the particles become increasingly rotationally symmetric (see Fig. 1 for an example of changing particle morphology).

For $r_0=0.33\,\mu$m a minimum in $C_{\mathrm {bak}}$ and correspondingly a maximum in $S_p$ can be observed for $f_{\mathrm {vol}}=0.55$. To investigate this feature the optical calculations were repeated for an increased resolution of $f_{\mathrm {vol}}$ with $\Delta f_{\mathrm {vol}}=0.05$. The results of these calculations are shown in Fig. 6. The panels show the different optical properties, the linear depolarization ratio (top left), the backscattering cross section (top right), the extinction cross section (bottom left), and the extinction-to-backscatter ratio (bottom right). The colors are as in Fig. 5. The shading represents the total range of values and in case of the properties averaged over all $N_c$ they represent the standard deviation. For an easier interpretation only the standard deviation over all $N_c$ is shown for $C_{\mathrm {bak}}$ and $S_p$. The morphology induced differences in $C_{\mathrm {ext}}$ are too small to be distinguishable. The higher resolution of $f_{\mathrm {vol}}$ reveals the minimum of $C_{\mathrm {bak}}$ and consequently the maximum of $S_p$ to be at $f_{\mathrm {vol}}=0.45$. The gradual change in $C_{\mathrm {bak}}$ and $S_p$ indicates that the behavior can likely be attributed to resonances in the backscattering cross section.

 

Fig. 6. Results of the coated convex polyhedra model for $r_0=.33\,\mu$m and $\Delta f_{\mathrm {vol}}=0.05$. The panels show the linear depolarization ratio (top left), the backscattering cross section (top right), the extinction cross section (bottom left), and the extinction-to-backscatter ratio (bottom right). The colors are as in Fig. 5. The shading represents the the total range and in case of the mean over $N_c$ the standard deviation. For $C_{\mathrm {bak}}$ and $S_p$ only the standard deviation is shown. For $C_{\mathrm {ext}}$ the respective ranges are too small to distinguish by eye.

Download Full Size | PDF

Finally, the particle grows with increasing coating thickness, i.e. decreasing $f_{\mathrm {vol}}$. The resulting increase in geometric cross section entails an increase of the optical cross sections $C_{\mathrm {bak}}$ and $C_{\mathrm {ext}}$. Thus we see that this morphologically realistic particle model yields optical properties that closely correspond to how one would expect dry and coated marine aerosol to vary with size and volume fraction.

4.2 Non-concentric spherical core shell

The results for the non-concentric spherical core shell model are displayed in Fig. 7. Here the rows represent the different values of $r_0$, with $r_0=0.33\,\mu$m in the top row, $r_0=0.67\,\mu$m in the center row, and $r_0=1.0\,\mu$m in the bottom row. The columns represent the different optical quantities, which are from left to right $\delta _l$, $C_{\mathrm {bak}}$, $C_{\mathrm {ext}}$, and $S_p$. The y-axis in each panel denotes the relative shift of the core’s center with respect to the shell’s center, with no shift at the top and maximum shift at the bottom. The x-axis represents the salt volume fraction $f_{\mathrm {vol}}$, which increases from left to right. Blank areas in the individual panels indicate the range of values for relative shifts inaccessible due to the constraint of having non-overlapping spheres.

 

Fig. 7. Optical properties at a wavelength of $\lambda =0.532\,\mu$m obtained by the non-concentrical spherical shell model. The columns correspond to the optical properties, these are from left to right: linear depolarization ratio, backsattering cross section, extinction cross section and extinction-to-backscatter ratio. The rows refer to the different initial sizes of the dry, uncoated particle, with $r_0=0.33\,\mu$m, $r_0=0.67\,\mu$m, and $r_0=1.0\,\mu$m in the top, center, and bottom row, respectively. The optical properties are shown as functions of the salt volume fraction $f_{\mathrm {vol}}$ and the relative shift.

Download Full Size | PDF

Along the y-axis the linear depolarization ratio behaves as expected. The larger the relative shift of the two spheres, the stronger the deviation from spherical symmetry, resulting in higher depolarization ratios. Along the x-axis two counteracting effects can be observed. The smaller $f_{\mathrm {vol}}$, the higher the water uptake. This gives the salt core more freedom to shift away from the center position, resulting in higher depolarization ratios. On the other hand, more water uptake also results in a shrinking of the salt core, as more solid material goes into solution. If the size of the inner sphere decreases beyond a certain point the solid core becomes too small to induce larger values of the depolarization ratio. For values of $f_{\mathrm {vol}}\approx 0.3$ the depolarization ratio reaches its maximum of $\sim 50\%$. This peak value is achieved at the maximum possible relative shift. For larger coating thickness, i.e. smaller values of $f_{\mathrm {vol}}$ the decrease in core size results in reduced depolarization.

The backscattering cross section displays characteristic oscillations with varying $f_{\mathrm {vol}}$, which is similar to the behavior we already noted in Fig. 9 for the reference geometry. For $r_{\mathrm {salt,dry}}$=1$\mu$m (bottom row), we can also discern corresponding oscillations along the y-axis. It is not surprising that a differential scattering property, such as $C_{\mathrm {bak}}$, exhibits such a high sensitivity to small changes in morphological properties.

The extinction cross section appears to be mainly controlled by the increase in particle size with decreasing $f_{\mathrm {vol}}$. For the extinction-to-backscatter ratio at $r_0=0.67\,\mu$m and $r_0=1.0\,\mu$m the highest values are found at $f_{vol}\approx 0.3$. In case of $r_0=0.67\,\mu$m this maximum appears to be caused by the minimum of the backscattering cross sections. For $r_0=1.0\,\mu$m a local minimum of $C_{\mathrm {bak}}$ coincides with the maximum of $C_{\mathrm {ext}}$.

Figure 8 shows depolarization ratio (top left panel), extinction-to-backscatter ratio (top right), extinction cross section (bottom left), and backscattering cross section (bottom right) obtained from the non-concentric spherical core shell model with $r_0=1.0\,\mu$m, and $f_{\mathrm {vol}}=0.45$ (blue line), $f_{\mathrm {vol}}=0.30$ (orange line), and $f_{\mathrm {vol}}=0.15$ (green line). As we already saw in Fig. 7 $\delta _l$ is largest for the highest relative shifts, i.e. for cores placed as closely as possible to the shell’s surface. $C_{\mathrm {ext}}$ is mainly controlled by the particle size, which grows with decreasing $f_{\mathrm {vol}}$. The increase and the decline in $C_{\mathrm {bak}}$ for $\mathrm {vol}=0.15$ and for $f_{\mathrm {vol}}=0.3$, respectively, for inclusions comparatively close to the shell’s surface, can be attributed to changes of the $F_{11}$-element in the backscattering direction, caused by resonances. By contrast, $C_{\mathrm {ext}}$, which is equal to $C_{\mathrm {sca}}$ (since the material is non-absorbing), displays only small changes for the range of relative shifts.

 

Fig. 8. Optical properties from the non-concentrical spherical shell model for $r_0=1.0\,\mu$m, and $f_{\mathrm {vol}}=0.45$ (blue line), $f_{\mathrm {vol}}=0.30$ (orange line), and $f_{\mathrm {vol}}=0.15$ (green line). The panels show the different optical properties - depolarization ratio (top left), extinction-to-backscatter ratio (top right), extinction cross section (bottom left), and backscattering cross section (bottom right).

Download Full Size | PDF

4.3 Superellipsoids and cube-sphere hybrids

Figures 9–11 show the results for the homogeneous superellipsoids (left column), the inhomogeneous superellipsoids (center column), and the cube-sphere hybrids (right column). In each individual panel $f_{\mathrm {vol}}$ increases from left to right, i.e., the water amount decreases from left to right, where the rightmost values correspond to dry particles. The shape changes are represented by the y-axis, with the bottom (with $n=0=\eta$) corresponding to a cube and the top (with $n=1=\eta$) corresponding to a sphere.

 

Fig. 9. Optical properties calculated with homogeneous superellipsoids (left column), coated superellipsoids (center column), and cube-sphere hybrids (right column) for $r_0=0.33\,\mu$m at a wavelength of $\lambda =0.532\,\mu$m. The rows show the different optical properties, as in Fig. 7. The red squares indicate the value and the position on the y-axis for the best fit.

Download Full Size | PDF

 

Fig. 10. As Fig. 9, but for $r_0=0.67\,\mu$m.

Download Full Size | PDF

 

Fig. 11. As Fig. 9, but for $r_0=1.0\,\mu$m.

Download Full Size | PDF

With increasing amount of water, i.e. decreasing $f_{\mathrm {vol}}$, $C_{\mathrm {ext}}$ increases. This can be attributed to the increase in size due to the added water. For $r_0$=0.33 and 0.67 $\mu$m, no significant dependence of $C_{\mathrm {ext}}$ on the rounding parameters $n$ or $\eta$ can be observed. For $r_0$=1.0 $\mu$m, only a very weak decrease in $C_{\mathrm {ext}}$ with increasing roundness can be seen.

For $r_0=0.33\,\mu$m, $\delta _l$ increases with decreasing $n$ and $\eta$, i.e. decreasing sphericity, and with decreasing $f_{\mathrm {vol}}$. For larger values of $r_0$, the dependence of $\delta _l$ on $f_{\mathrm {vol}}$ and $n$ displays an increasingly complex pattern of resonance features. But for all sizes, volume fractions, and model particles, $\delta _l$ converges to zero as the particles approach spherical shape ($n=\eta =1$). Note, that the values of $\delta _l\geq 0.45$ for the inhomogeneous superellipsoids at $r_0=0.67\,\mu$m and $r_0=1.0\,\mu$m extend to $\delta _l=0.6$.

The backscattering cross section displays distinct size-dependent (i.e., $f_{\mathrm {vol}}$-dependent) resonances that increase in amplitude as the particles approach spherical shape ($n,\eta \to 1$). For nonspherical, cube-like particles ($n,\eta \to 0$) such resonances are quenched by taking an ensemble average over random orientations. This effect is particularly evident for $r_0$=0.67 $\mu$m (Fig. 10). These resonances also affect the extinction-to-backscatter ratio, which is composed of the extinction and the backscattering cross sections. Local minima of $C_{\mathrm {bak}}$ generally coincide with local maxima of $S_p$ and vice versa.

A comparison of the three columns in Figs. 9–11 allows us to identify similarities and differences among the three classes of model particles. First, we see that differences become more pronounced with increasing $r_0$, which was expected. Second, we find that homogeneous superellipsoids and cube-sphere hybrids display a lot more similarities in optical properties than one may have expected. This indicates that the change in overall shape has a much more profound impact on the optical properties, especially on depolarization, than the local rounding of the particles’ edges. The change in overall shape is similar in both models, while the edge-rounding effect is present in superellipsoids and absent in the cube-sphere hybrids. Third, we observe that differences between homogeneous and inhomogeneous superellipsoids (columns 1 and 2) are more distinct than corresponding differences between homogeneous superellipsoids and cube-sphere hybrids (columns 1 and 3). This observation is valid for $\delta _l$, $C_{\mathrm {bak}}$, and it is particularly pronounced for $S_p$. This somewhat unexpected result shows that, despite the relatively low optical contrast between the salt core and the brine coating, inhomogeneity can have a distinct effect on lidar-relevant optical properties.

The inhomogeneous superellipsoids can yield higher values of $\delta _l$, compared to homogeneous superellipsoids. This confirms the results reported by [28].

For each value of $f_{\mathrm {vol}}$ and $r_0$ the values of $n$ and $\eta$ best replicating the mean values of $\delta _l$, $C_{\mathrm {bak}}$ and $C_{\mathrm {ext}}$, as obtained from the coated convex polyhedra model, are identified. For this the geometric mean error is introduced:

For each optical property $Q_k$ the relative differences between the coated polyhedra model as reference model and the respective simplified model, as indicated by the superscripts, are calculated. The sum extends over the three optical properties $Q_k$=$\delta _l$, $C_{\mathrm {bak}}$, and $C_{\mathrm {ext}}$. (The lidar ratio is not included in the summation, because it is merely a quantity derived from $C_{\mathrm {bak}}$ and $C_{\mathrm {ext}}$.) The best fit is obtained by choosing for each value of $f_{\mathrm {vol}}$ and $r_0$ the superellipsoid’s roundness parameter $n$ or in case of the cube-sphere hybrids the value of $\eta$, for which $\Delta$ has a minimum.

In Figs. 9–11 the roundness parameter $n$ or in case of the cube-sphere hybrid $\eta$ minimising $\Delta$ is indicated by a red square. Generally, with decreasing $f_{\mathrm {vol}}$ the superellipsoids and cube-sphere hybrids giving the best fit become increasingly spherical. This reflects the increasingly spherical shape of the reference model particles (see Fig. 3). Thus, despite substantial morphological simplifications in all three classes of model particles, we find a correspondence in overall shape between the morphologically complex reference particles and the highly symmetric model particles.

Figure 12 compares the mean coated convex polyhedra model results as a reference (blue) with the best fits of the inhomogeneous (green), homogeneous (orange) superellipsoids, and the cube-sphere hybrid (red). The extinction-to-backscatter ratio was not included in the calculation of the best fit, as this would implicitly weigh $C_{\mathrm {ext}}$ and $C_{\mathrm {bak}}$ stronger than $\delta _l$. The best fit of the inhomogeneous superellipsoids result in the smallest deviations from the coated convex polyhedra model for $C_{\mathrm {ext}}$ at $r_0=0.33\,\mu$m and $r_0=0.67\,\mu$m, as well as for $\delta _l$ at $r_0=0.33\,\mu$m and $r_0=1.0\,\mu$m. For $C_{\mathrm {bak}}$, $C_{\mathrm {ext}}$, and $S_p$ at $r_0=1.0\,\mu$m, and $S_p$ at $r_0=0.67\,\mu$m the best fit of the cube-sphere hybrid model results in the smallest deviations from the reference model. For the remaining quantities the homogeneous superellipsoidal model results in the smallest deviations. Thus, none of the three models sticks out as significantly superior to the other two. The discrepancies in backscattering and extinction cross section, despite the overall consistent trends, between the reference and the best fits of the morphologically simpler models, result in rather poor agreement between the models with respect to the extinction-to-backscatter ratio. The differences are especially apparent for $r_0=0.67\,\mu$m; here $S_p$ obtained with the reference model decreases with increasing $f_{\mathrm {vol}}$, whereas, $S_p$ for the best fit of the simplified models increase. For applications with less emphasis on the depolarization ratios these differences could be reduced by increasing the weight on the optical cross sections, or by including the lidar ratio in the error metric in Eq. (9).

 

Fig. 12. Comparison of $\delta _l$, $C_{\mathrm {bak}}$, $C_{\mathrm {ext}}$, and $S_p$ (columns from left to right) for $r_0=0.33\,\mu$m, (top row) $r_0=0.67\,\mu$m (middle row), and $r_0=1.0\,\mu$m (bottom row). Blue lines show the arithmetic mean of the reference model. The error bars denote the standard deviation. Orange, green, and red indicate the best fit for homogeneous (denoted as hom. s. ell. in the legend), inhomogeneous superellipsoids (referred to as inhom. s. ell.), and cube-sphere hybrids (denoted as csh) respectively.

Download Full Size | PDF

Examples for the geometries giving the best fit for $r_0=0.67\,\mu$m and $f_{\mathrm {vol}}=0.55$ are shown in Fig. 13. Furthest left is the geometry of coated polyhedra (with $N_c=50$), which gives $\delta _l$ and $C_{\mathrm {bak}}$ values closest to the mean value. Thus, this specific geometric realization is optically most representative for the ensemble consisting of 15 different realizations (three values of $N_c$ with five different stochastic realizations each). The remaining three geometries are, from left to right - a homogeneous superellipsoid (with $n=0.4$), an inhomogeneous superellipsoid (with $n=0.7$), and a cube-sphere hybrid (with $\eta =0.4$).

 

Fig. 13. Coated polyhedron with $r_0=0.67\,\mu$m, $f_{\mathrm {vol}}=0.55$, and $N_c=50$ (left), which gives values of $\delta _l$ and $C_{\mathrm {bak}}$ closest to the mean, shown in Fig. 12. The other model particles show the respective realization giving the best fit. From left to right these are - a homogeneous superellipsoid ($n=0.4$), an inhomogeneous superellipsoid ($n=0.7$), and a cube-sphere hybrid ($\eta =0.4$).

Download Full Size | PDF

The non-zero depolarization ratio of the reference geometry shown here stems entirely from the nonspherical salt-core, as the watery coating is already spherical. The simplified geometries indicate, that the overall non-sphericity of the superellipsoids and the cube-sphere hybrids are not directly mirrored by the respective coated polyhedra. However, the inhomogeneous superellipsoid is closer to a spherical form, than the other two geometries. The best-fit roundness parameters of the inhomogeneous superellipsoidals, as can be seen in Figs. 9–11, are always equal to or larger than those of the homogeneous superellipsoids, i.e. $n_{\mathrm {inhomogeneous}}\geq n_{\mathrm {homogeneous}}$. Thus, effects from centered inhomogeneities in the inhomogeneous superellipsoids can be compensated by less spherical homogeneous superellipsoids. This is a remarkable result, as inhomogeneity can, in other cases, have such a profound effect on optical properties that attempts to mimic their effect by use of homogeneous model particles have failed. For instance, for particles with a refractive index of $m=1.5 + 0.002$i (representative of silicate particles) homogeneous spheroids are capable of reproducing optical properties of other homogeneous particles of various shapes, including Gaussian random spheres and cubes. However, attempts to mimic the optical properties of inhomogeneous spheroids with multiple spherical air inclusions by use of homogeneous spheroids were unsuccessful, even when trying different effective medium approaches, different metrics for the equivalent size of homogeneous and inhomogeneous particles, and when fitting a shape-distribution of the homogeneous model particles to the inhomogeneous reference particles [63]. In our case, inhomogeneity still has a significant effect, which is consistent with what has been previously seen for mineral dust [63,64]. However, the effect is not so profound that the optical properties cannot be mimicked by use of adequately selected homogeneous model particles. A plausible explanation is that in our study the optical contrast between the brine coating and the salt core is relatively small, which moderates the impact of inhomogeneity on optical properties.

5. Discussion and conclusion

The first question we formulated in the introduction concerned the performance of a detailed reference model that describes the water-uptake process by use of physically plausible assumptions. Qualitatively, the model largely behaves as expected. The depolarization ratio converges to zero as more water is added, resulting in a spherical geometry. At the same time the optical cross sections increase due to particle growth.

More quantitatively, we found that the proposed coated polyhedra model covers mean values of $\delta _l$ ranging from 0 to 0.35. The optical cross sections in this study, which are more directly linked with the particle size than $\delta _l$, cover ranges in case of $C_{\mathrm {bak}}$ from 0.025 to $2.6\,\mu$m$^{2}\,$sr$^{-1}$, and in case of $C_{\mathrm {ext}}$ from 1.4 to $15.5\,\mu$m$^2$. The lidar ratio $S_p$ ranges between 5 and $90\,$sr. It must be emphasized that these are results for individual particles, which cannot be directly compared to reported lidar field observations, as presented in Tab. 1, since these refer to particle ensembles averaged over a size distribution. Nevertheless, we can attempt to make a first assessment whether or not size-averaged optical properties are likely to fall within the range of reported measurements.

Table 1. Reported values of $\delta _l$ and $S_p$ at $\lambda = 0.532\,\mu$m from lidar field measurements of marine aerosol, including the reported values of RH. The values of $S_p$ from [65] (*) were assumed to be $S_p=20\,$sr, based on values from [66]. In [67] values of $\delta _l <5\%$ were reported for a wide range of RH (**), without giving the explicit range. *** marks mean values of $S_p$ reported for the entire case study.

View Table

There are several reports of lidar field measurements, as well as two studies that investigated satellite-borne lidar measurements of marine aerosol over a larger spatial and temporal scale [74,75]. In [74] a global mean value of $S_p$ for marine aerosol was reported as $S_p=26\,$sr. This study was based on night-time observations by the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) satellite sensor between December 2007 and November 2010. $S_p$ was found to depend on the surface wind speed $u$ and ranges between $S_p=22\pm 7\,$sr ($u>15\,$m$\,$s$^{-1}$) and $32\pm 7\,$sr ($u<4\,$m$\,$s${-1}$). Another study [75] used night-time data measured by the CALIOP instrument during the austral winter in a ten year (2007-2016) time period over the Southern Ocean (40$^{\circ }$S-65$^{\circ }$S). In [75] $\delta _l$ for clean marine aerosol, as identified in the CALIOP product, was found to range between 6% (RH$=100\%$) and 17% (RH$=0\%$).

Summarizing the reported observations in Table 1, the linear depolarization ratio of marine aerosol covers a range between $<1\%$ and $25-29\%$, although most values are below $20\%$. The reported measured extinction-to-backscatter ratio varies between $10\pm 2\,$sr and $32\pm 17$, but $S_p=10\pm 2\,$sr [73] and $S_p=13\pm 3\,$sr [72] appear to be low outliers.

In many cases, our reference model falls within the range of reported measurements, but at some individual sizes the model exceeds the measurements. For example, the values of $S_p$ at $r_0=0.33\,\mu$m and $\delta _l$ at $r_0=1.0\,\mu$m, exceeds the range of reported observations. However, in [19] we reported for small, individual superellipsoids ($r_0=0.25\,\mu$m, $m=1.5484+i0$) values of $S_p \sim 100\,$sr, which are clearly larger than the observation-derived lidar ratios. But after size-averaging we obtained a range of $S_p=12-20\,$sr. This indicates that size-averaging is likely to quench the extreme values observed for some isolated particle sizes, and to yield ensemble-averaged values of $S_p$ and $\delta _l$ more in line with the observations.

Further, the coated polyhedra model can be fine-tuned by changing the number of randomly placed points $N_c$, thus affecting the salt core’s initial shape or by tuning the surface potential calculations, affecting the shape of the coating and of the partially dissolved salt core. Future guidance would need to be provided by laboratory studies.

We can conclude that this prospecting study shows sufficient promise to warrant further investigation of size-averaged optical properties computed with our reference model. Owing to the high computational requirements, this will be a computationally very demanding follow-up study.

The second question we formulated in the introduction was whether or not the particle can still depolarize once the core and the shell have reached spherical shape, owing to a variation of the positioning of the salt core within the brine mantel. The results of the non-concentric spherical shell model suggest that considerable values of the linear depolarization ratio (up to $\sim 0.5$) can be induced by off-center inhomogeneities, especially for particles with $f_{\mathrm {vol}}<0.45$. This is of comparable magnitude as the effect that a nonspherical coating can have on depolarization. This is a very unexpected result. We know from studies of coated soot aerosols that the positioning of the soot core within the mantle has a rather modest effect on the depolarization ratio (e.g. [76]). While coated soot particles are characterized by a high optical contrast between the soot core and the coating, our brine-coated salt particles are composed of optically very similar materials. It is entirely unexpected that inhomogeneity in a composite particle of such low optical contrast can have such a profound impact on depolarization. However, we emphasize that we obtained the largest values of $\delta _l$ for individual sizes and volume fractions. Size-averaging is likely to significantly moderate the effect of core positioning on depolarization. We note also that we currently know very little about the positioning of the salt core within the brine mantle. Our investigation was motivated by energy potential calculations presented in [39], which suggest that it is more likely for the core to be positioned off-center close to the surface of the coating. But further measurements are required to determine if and how frequently such off-center inhomogeneities can be observed in partially dissolved sea-salt crystals.

The third question concerned the capacity of simplified model particles to reproduce the optical properties of the reference model. We found that both superellipsoidal models and the cube-sphere hybrid model cover a large range of optical properties. By changing the respective model particle’s non-sphericity, quantified either by $n$ in case of superellipsoids or by $\eta$ in case of cube-sphere hybrids, the optical properties can be fitted to different stages of the partial dissolution process of sea salt particles, as described by the coated polyhedra model. All three models can reproduce the optical properties comparatively well, with the homogeneous superellipsoidal model resulting in the smallest deviations from the convex polyhedra model in five of twelve cases, as defined by optical properties and $r_0$. The lidar ratio $S_p$ was most challenging to reproduce. However, a likely explanation is that $S_p$ was not explicitly accounted for in the error metric in Eq. (9) that was employed in determining the best-fit model particles.

Th final question was if we can gain some insight into the relevance of different morphological features for the optical properties. We found that, despite the low optical contrast between the salt core and the brine mantle, inhomogeneity can have a strong impact on depolarization. However, we also found that both homogeneous superellipsoids and cube-sphere hybrids can compensate for inhomogeneities. In fact, the overall nonsphericity of the coating seems to be a key morphological parameter that impacts the depolarization ratio, but also the backscattering cross section. Further, a comparison between the homogeneous superellipsoidal model and the cube-sphere hybrids revealed that both models give very similar results. The main morphological difference between the two models is that the former accounts for rounding of the particle edges, while the latter does not. This indicates that the overall shape has a significantly more prominent impact on the optical properties than rounding of edges and corners.

One important question that we did not address is the sensitivity to the refractive index of sea salt. For dry marine aerosols we used the refractive index of sodium chloride. Under realistic conditions, organic and inorganic impurities can modulate the refractive index of sea-salt particles. A recent study based on superellipsoids [19] showed that a decrease in the real part and an increase in the imaginary part, which would be typical for organic impurities, can lower the backscattering cross section and raise the lidar ratio. The depolarization ratio will mainly increase with decreasing real part, and decrease with increasing imaginary part of the refractive index.

Further refinements of the coated polyhedra and superellipsoid models are conceivable, e.g., by considering non-unity aspect ratios of the superellipsoids, i.e. $a=b\neq c$ or $a\neq b \neq c$. The search for suitable simplified models could be widened accordingly. However, development of such highly tunable models with several free parameters will have to rely on guidance provided by laboratory measurements, which are currently in short supply.