The 2-D video disdrometer (2DVD) was developed by Joanneum Research (Graz, Austria) to detect single raindrop particles, and the instrument has been modified to cover the errors caused by turbulence effects (Nešpor et al., 2000). The device is able to observe the shape, VT, and β of a single particle using optical light. The ability to analyze a single particle is a significant advantage compared to other disdrometers, such as the Joss–Waldvogel disdrometer (Joss and Waldvogel, 1967), the Precipitation Occurrence Sensor System (Sheppard, 1990), and Parsivel (Löffler-Mang and Joss, 2000). For instance, Parsivel considers a fixed measurement area without any consideration of particle shape (e.g., Tokay et al., 2014), while 2DVD observes particles by passing them through a 100 cm2 observation area consisting of two light sources, i.e., reflecting mirrors and two cameras, with one camera set 6.2 mm above the other and collecting data with a resolution of 630 pixels; this results in a pixel size of 0.2 mm at 55 kHz (Kruger and Krajewski, 2002). Particles passing through the observation area yield shape information according to the radiation intensity of the light sources, which is helpful for calculation of γ and β. The VT of particles is calculated using the height difference between the two cameras. Based on these advantages, the oscillation and particle shape of raindrops can be analyzed by 2DVD (Thurai and Bringi, 2005). Böhm (1989) analyzed the aerodynamic properties of an irregular hydrometeor and Huang et al. (2010, 2015) used 2DVD to analyze the features of irregularly shaped snow. There have been few previous aerodynamic analyses of volcanic ash particles performed using 2DVD, which is able to detect and analyze volcanic ash particles with a range of irregular shapes. Thus, 2DVD offers a unique approach as a new observation strategy.

Volcanic ash particles have various shapes that can be detected by 2DVD (Fig. 1). In the case of raindrops, the drop size distribution (DSD) is dependent upon the break-up and coalescence processes occurring via up and downdrafts, since the forces of gravity and buoyancy can easily affect raindrop shapes (Rosenfeld and Ulbrich, 2003). However, solid particles do not readily change shape when falling without the influence of forces such as collision. It is thus inferred that many particle shapes would be found in the atmosphere, and that it would be possible to define and classify each particle shape type if we were able to accurately detect a single particle. Thus, the range of γ for solid particles would be expected to be wide compared to that of raindrops, and various values of VT and β would likely be observed. The γx of a particle is defined as the ratio of height to width for the observation direction x, and its representative value is calculated using the geometric means of the two γ (γ1, γ2) detected by cameras 1 and 2, respectively (Eq. 1): 1γ1(2)=Height1(2)Width1(2),γ=γ1γ2.

The difference in angle between the rotating symmetric axis and vertical axis is defined as β. The counter-clockwise (clockwise) movement of the rotating symmetric axis has a positive (negative) value and the entire range is 180∘ (from -90 to 90∘) with 0∘ as the center.

It is necessary to consider the true axis ratio (γT) to correctly define the particle shape (Fig. 2). The apparent axis ratio (γA) considers the effect of β but the γT does not. The 2-D coordinates (x,z) of the particle shape with β are defined as follows: xA=rcos⁡(θ+β),zA=rsin⁡(θ+β),xT=rcos⁡θ,zT=rsin⁡θ, where subscript A is the coordinate of the original data coordinate considering the β and subscript T is the modified data coordinate. The symbol r refers to the length from the data point to the center and the symbol θ represents the degrees of data coordinates from the positive x axis, which range between 0 and 180∘. In this paper, γ stands for γA for convenience.

An objective criterion for particle shape type was considered since particle shapes can be highly diverse and irregular (e.g., Bagheri and Bonadonna, 2016; Dioguardi et al., 2017, 2018; Garboczi and Bullard, 2017). In the case of irregular particles, the γ can change according to the observation direction; however, any criterion should be able to define the particle shape types strictly and reliably. To solve this problem, particles with a rotating symmetric axis were the main target of the present study. Therefore, we considered oblate spheroid (O), prolate spheroid (P), and sphere (Sp), which all have a rotating symmetric axes. Among these particle types, the major axes of the oblate and prolate spheroids could be horizontally (H) and vertically (V) oriented with respect to the ground, respectively. Thus, the various particle shapes were divided into five types as follows; oblate spheroid with horizontal rotating axis (OH), oblate spheroid with vertical axis (OV), prolate spheroid with horizontal rotating axis (PH), prolate spheroid with vertical rotating axis (PV), and Sp.

To define these particle shape types, a strict definition of the γT is required, which can be calculated from the β. As with the γ, the two β values are automatically calculated by 2DVD. In the case where the β is assumed to be 0∘, the rotating symmetric axis for OH and PV can be defined, since it is observed for any observation direction parallel to the ground. However, in the case of OV and PH particles, the rotating symmetric axis cannot be defined when the observation direction is parallel. In the case where the β is not 0∘, γT for all particle shape types would not change when oscillation occurs in a direction orthogonal to the observation direction, but it is difficult to estimate both γT and β when particle oscillation appears in a direction parallel to the observation direction. The ability to restore the γT and β relative to this observation direction is limited, which is one of the main disadvantages of the 2-D observation strategy.

Based on these facts, a major β was selected based on the following reasoning: (i) a β for the observation direction with lower (higher) γT for OH (PV) is selected. (ii) In the case of OV (PH), for which the rotating symmetric axis was observed for only one observation direction, β was considered where the value of β had a higher (lower) γT than that of the other observation direction. Therefore, β with a lower (higher) γT in two observation directions for the case of an oblate (prolate) particle was considered a meaningful value. The perfect sphere could not have their value of β determined theoretically because there is the possibility of a rotating symmetric axis in any direction.

Based on the definition of β (Fig. 2), the perfect condition with respect to ellipsoids is satisfied when |β|=0∘ (90∘) for OH and PV (OV and PH); these values are defined as the center values. However, 2DVD calculated that the β for each particle shape type was concentrated around |β|=0∘ (90∘) with respect to horizontally (vertically) oriented particles, which correspond to OH and PH (OV and PV). Furthermore, analysis of particles with an orthogonal center angle from 0∘ is difficult, since they have two center angles (±90∘). To address observation errors and enhance the convenience of analysis, all center angles were set to |β|=0∘ and modified to give the representative canting angle, βR, using the following equation: βR=β-β0, where β0 is the orienting angle, defined by the central angle of oscillation. In the case of vertically oriented particles (OV and PV), β0 could be defined as ±90∘. The sign of β0 follows that of β.

After removing β, each particle shape was defined using γT (Table 1). Note that a 10 % bias range was allowed, to take observational error into account. For example, a particle was considered a sphere when 0.9<γT<1.1, which is an applied 10 % bias range from γT=1. In addition, the particle types OH and PV (OV and PH) were classified when the value of |γ1–γ2| was smaller (larger) than 0.1γT, to consider particles with only a rotating symmetric axis.

The VT of volcanic ash is required to estimate the RA (kg m-2 s-1) on the ground where this depends on atmospheric density (ρg in g cm-3), T, Re, CD, D, ρs, and its shape. Kunii and Levenspiel (1969) developed a theoretical VT equation: 2VT=4ρs-ρggD3ρgCD0.5(100<Re<104).

Later, Suzuki (1983) developed a theoretical VT equation for tephra. Bonadonna et al. (2011) then modified the theoretical VT equation suggested by Kunii and Levenspiel (1969) with observed ash data, which implied that the result of the theoretical VT equation could be unsuitable for nonspherical particles. Based on these equations, various CD equations considering nonspherical particles were subsequently developed. Tran-Cong et al. (2004) developed a new equation for CD using the function of circularity, and Hölzer and Sommerfeld (2008) introduced a progressed CD equation considering two types of sphericity: lengthwise (Φ||) and crosswise (Φ⊥). This equation is as follows: 3CD=8Re1Φ||+16Re1Φ+3Re1Φ3/4+0.42×100.41(-log⁡(Φ))0.21Φ⊥.

The Re is defined as follows: 4Re=ρgVTDμ, where μ is the dynamic viscosity (kg m-1 s-1), which we assumed to be 1.983×10-5 based on atmospheric conditions at a T of 25 ∘C. Three types of sphericity were defined as follows: 5Φ=πD2SA, where SA is the surface area of the particle (mm2). The lengthwise sphericity is defined as the ratio between the cross-sectional area of the volume-equivalent sphere and the difference between half the surface area and the mean of the projected vertical cross-sectional area (AV) of the particle (Eq. 6): 6Φ||=πD24(0.5×SA-AV).

The crosswise sphericity is the same as the lengthwise sphericity, except for the denominator, which includes the projected horizontal cross-sectional area of the particle (AH), defined as follows: 7Φ⊥=πD24AH.

It is noteworthy that the VT is required to calculate the Re and VT, which refers to the final product. To solve this problem, the theoretical VT (Eq. 2) was used as the input value of Eq. (4) until Eq. (2) converged.

Japan has around 10 % (110) of all of the active volcanos in the world. Sakurajima (1117 m.a.s.l, 31.58∘ N, 130.65∘ E, Kyushu, Japan) is an active volcanic island formed around 13 000 yr ago, and its tephra is approximately 60 %–66 % SiO2 Peléan type (Oguchi et al., 2009; Takahashi et al., 2013). The major eruptive events in the historic age of Sakurajima were 1471–1476 (Bunmei era), 1779–1782 (An'ei era), and 1914 (Taishō era). Sakurajima is an andesitic volcano with two peaks (Kita-dake and Minami-dake). Volcanic activity at Kita-dake ended around 4900 yr ago when it changed to Minami-dake. Activity has centered on Showa crater from 2006 (Iguchi, 2013). Showa crater is located on the eastern flank, approximately 500 m east of Minami-dake (southern peak) on Sakurajima. It was appeared in 1939 after 1 month of eruptions (Yokoo and Ishihara, 2007). The Minami-dake summit crater was the only active center of Sakurajima until the recommencement of Showa crater from 1948 to 2006. The eruptive activity of Showa crater resumed in June 2006, and vulcanian eruptions gradually increased in the autumn of 2009 (Hotta et al., 2016). The Japan Meteorological Agency (JMA) reported that the eruption frequency of Sakurajima would increase significantly from 2009, and the accumulated ash fall exceeded 3.5 kg m-2 in Kagoshima in 2012. The Ministry of Land, Infrastructure, Transport, and Tourism (MLITT) installed an operational X-band radar 10.7 km from the vent, as well as 16 automatic volcanic ash weight measurements, to observe volcanic eruptions in 2011 (Fig. 3).

The data were collected by automatic volcanic ash weight measurements performed on Sakurajima (Tajima et al., 2015; Maki et al., 2014, 2016). The free-fall experiments were divided into two types: one was performed for each phi scale (Φ=-log⁡2D) from Φ=3 to -4 (0.125<D<16 mm) and the other was not considered on a particle-size scale. The former data, expressed by A and B (Type 1), were collected at two sites and screened by size (Fig. 4); the latter data, expressed as C–E (Type 2), were collected at 18 sites (Table 2). Free-fall experiments on collected volcanic ash particles were carried out in the large-scale rainfall simulator of the National Research Institute for Earth Science and Disaster Prevention (NIED) in Tsukuba, Japan. The collected particles were dropped manually around 17 m from the ground and re-collected by a third-generation 2DVD (Maki et al., 2016). Each sample was dropped for 30 s to stimulate dispersion, and the measurement period was 1 min. To avoid wind effects including turbulence, the 2DVD was surrounded by a 3 m3 wind-breaking wall (Fig. 5).

The free-fall experiments were conducted at intervals of 1 min over 6.5 h, as shown in Fig. 6. The number of particles detected by 2DVD was less than 10 000 for 1 min, and the particle size range of the Type 1 data set was proportional to its phi scale, since small particles may be contained by screening.

Figure 7 shows the distribution of raw data (the number of data: 274 215) for VT and γ with D. There were various γ from 0 to 2 when D<2 mm, and most of the data were concentrated near γ=0. The γ values converged around 1 and their distributional range decreased with D. The median value with a 0.25 mm D interval corresponded well to the center of the data contour. The median line converged around γ=0.935 based on the correlation coefficient value (CC). When this was higher than 0.95 for each D interval, the data converged. According to this condition, the range of 2<D<5 mm was satisfied and the mean value was calculated using these data. The VT had a wider range when D<2 mm but the median line corresponded well to the center of the data (Fig. 7b). The line representing the largest number of data is lower than the volcanic ash discussed by Bonadonna et al. (2011). To select a available range of D, a theoretical terminal velocity equation (VT,Ref) for a non-sphere corresponding to Eqs. (2)–(7) was used as the reference. The particle density associated with the eruption of Sakurajima is between 2.43 and 2.59 g cm-3 (Oguchi et al., 2009), but the actual particles contain air vacuoles (Van Eaton et al., 2012). This means that ρs including vacuoles is smaller than ρs alone. Therefore, the minimum ρs was considered to be 2.43 g cm-3, and this was used as an input parameter. The atmospheric conditions of T and P were considered from an automatic weather station (AWS), supported by the JMA. The falling height of a particle, which followed the aforementioned conditions (γ=0.935, ρs=2.43 g cm-3) when D=4 was lower than that under the condition of the free-fall experiment (17 m) and it reached 90 % of VT (13.9 m); therefore, the available data range is considered to be D≤4 mm, and this would satisfy the terminal fall velocity. The detailed equations used in the present study are shown in Appendix A.

The 2DVD was originally developed to detect raindrop hydrometeors. For this reason, additional quality control (QC) checks were deemed necessary to ensure applicability to non-hydrometeors, such as volcanic ash particles. Specifically, we performed the following three QC procedures for accurate analysis of the data. i.

Particle D>0.25 mm was selected in consideration of the minimum spatial resolution of 2DVD.

The ash particle size over the entire volcanic ash sample was skewed leftward, and the dominant particle shape type changed with particle size (Fig. 8). The particles were predominantly horizontally oriented (75.51 %) and vertically oriented (21.60 %). Oblate and prolate spheroids made up 76.26 % and 23.85 % of the particles, respectively. Hence, the particles were mainly OH (57.38 %) or PH (15.88 %) (Table 2). The particles were predominantly 0.25<D<0.5 mm (63.00 %) or 0.5<D<1 mm (32.80 %). Relatively few particles had D>1 mm (4.20 %).

There was large variation in shape among particles 0.25<D<0.5 mm, but the variation decreased with increasing D. All of the particle shape types had the largest number of particle at D>1.0 mm and the next largest were shown at 0.5<D<1.0 mm, except for OH. In total, 95.80 % of OH particles had D<1 mm, at D<0.5 mm the value was 75.68 %, and at D=0.5–1 mm it was 22.36 %. In the cases of PH and OV, 93.63 % and 93.87 % of these particles, respectively, had D<1 mm. Beyond D>1 mm, the differences in the number of particles for each particle shape type were considerably decreased.

The VT for the entire particle shape types follows a polynomial regression analysis that was applied to define the nonlinear relationship between VT and D (Fig. 9a). It corresponds to that obtained by Miwa et al. (2015), who analyzed Parsivel data using the same laboratory experiments. The inflection point of VT was at D<1.3 mm and it came from an increase in the number of OH and a decrease in their VT when D<1 mm. As a result, the VT relationship for each particle shape type can be seen clearly.

The observed values of VT were well classified by particle shapes (Fig. 9b). The highest values of VT were recorded in the following order: prolate, sphere, and oblate. Vertically oriented particles had higher VT values than horizontal ones. The VT for every particle type can be expressed in a power law form, except for OH. OH particles followed the regression line relatively closely and showed the highest CC and root-mean-square error (RMSE) values of 0.94 and 0.46 m s-1, respectively. Horizontally oriented particles had relatively high correlations (OH: 0.94; PH: 0.87) compared to those with a vertical orientation (OV: 0.75; PV: 0.71). The VT relationships and those of statistical parameters are summarized in Table 3.

To verify the reliability of the particle data obtained by 2DVD, which was originally developed to detect liquid raindrops, ρs, CD, and Re, as well as theoretical VT values according to these parameters, were analyzed. To calculate the parameters of interest, including the surface area and cross-sectional area of irregular particles, we applied the irregular particle volume estimation equations of Huang et al. (2010).

Particle densities were estimated using the VT,Ref, converged to 2.37 g cm-3 when D<1.5 mm (Fig. 10a). This value corresponds well to that of the minimum ρs (2.43 g cm-3) reported by Oguchi et al. (2009). The slight difference is likely due to observation errors and the presence of vesicles (e.g., Seligman et al., 2016). The median value of ρs is changed to 0.5<D<1.5 mm, and this range corresponds to that of VT. Horizontally oriented particles (OH and PH) have relatively smaller ρs and vertically oriented particles (OV and PV) have higher density (Fig. 10b); spheres have particle densities that accord best with D. The median ρs values for all particle shapes converged when D<2 mm, and the converged ρs values ranged from 2.35 to 2.50 g cm-3.

The Re and CD for the all particle shapes ranged from 10 to 4000 and 0.6 to 20, respectively (Fig. 11a). Higher values of CD were observed when log(Re)<1.845 (Re<70); above this threshold, CD dramatically decreased. These results were derived according to the number of OH particles, which mainly had D<0.5 mm and had higher CD and lower Re. Particle shape was divided into two types: OH and others (Fig. 11b). OH particles had higher CD compared to the other particle shapes, which in turn showed few differences among themselves. The OH particles experience strong drag forces under the same flow conditions, leading to lower VT. The differences between OH and other particles diminished with Re<1000. The other particle shapes had relatively higher CD in the range 10<Re<3000. Rong et al. (2015) analyzed the relationship between Re and CD for oblate and prolate particles and showed that OH particles had higher CD compared to the reference line (Clift and Gauven, 1971) in the range 0<Re<400. Each relationship is summarized in Fig. 11.

The γ of a particle affects the backscattering power of electromagnetic waves and is necessary to calculate the horizontal reflectivity (Z in dBZ), differential reflectivity (ZDR in dB), and specific differential phase shift (KDP in ∘ km-1). Note that previous studies have analyzed the γ distribution for raindrops and snow, including hail; however, few studies have been reported on γ of ash particles compared to those for hydrometeors.

Figure 12 shows the quartiles and median values of γ for all particle shapes and for each individual particle shape type. The γ had a higher standard deviation (σγ) of >0.25 when D<0.75 mm, which decreased and converged to σγ=0.15 when D>1 mm (Fig. 12a). The γ in the lower σγ range converged to γ=0.94 and could be expressed as follows: 8γ(D)=0.94-0.25exp⁡(-1.90D).

The particles are more easily classified by shape than by VT (Fig. 12b). Every particle shape type was independent of D, except for OH. The γ of each particle type is expressed via the following relationships (Eqs. 9–12). 9γDOH=0.37tanh1.84D-1.88+0.3810γOV=1.1511γPH=0.8812γPV=1.24

The parameter γ(D)OH was calculated using the hyperbolic tangent (tanh) for the following reasons: (i) its range of values was wider than those of other particle types, (ii) its data distribution changed continuously with D, and (iii) it was present in a higher proportion (30.44 %) compared to the other parameters. We found that variations in γ decreased with D and the proportion of Sp shapes increased when D>2 mm. The particle types OV and PH showed a wide distribution over 1<γ<1.5 and 0.4<γ<1.0, respectively, when D<2 mm, but the variability in median values was relatively low. The relationships of γ for each particle shape type could be expressed by constant values at γ=0.75 (OH), 0.88 (PH), 1.15 (OV), and 1.24 (PV), respectively, and these differences are around γ=0.12.

Statistical analysis of β is required to understand the aerodynamic properties of volcanic ash particles and the input parameters of T-matrix scattering simulations to verify the observed radar variables. The histogram of βR is shown in Fig. 13.

More than 95 % of β values for each particle shape type were concentrated in the range |βR|≤30∘, with 0∘ as the center (Fig. 13). The particles were symmetrically distributed around 0∘ and more than 50 % were concentrated in the range |βR|<4∘. The horizontally oriented β distribution was relatively narrow (|βR|<20∘) and exhibited a unimodal distribution. It is noteworthy that 90 % of OH particles were concentrated in the range |βR|<4∘. The vertically oriented β distribution was relatively broad (|βR|<30∘) and followed a bimodal distribution. PV exhibited a bimodal form, but this was not symmetrical around 0∘. For spheres, the β distribution was narrower, similar to horizontally oriented particles, and it had a bimodal distribution, similar to vertically oriented particles, indicating that the independent features of both orientations were combined.

The values of |βR‾| and |σβ| for OH and PV (OV, PH) were 0 and 3.5∘ (0.4, 13.1∘) and 1.3 and 12.7∘ (0.2, 10.9∘), respectively. OV (OH) had the highest (lowest) value of |σβ|. This validates the assumption of Marzano et al. (2012) under stable conditions (|σβ|=10∘). Therefore, we believe that the tumbling phenomenon (|σβ|>30∘) of the particles under calm atmospheric conditions was likely to be minor.

To analyze the correlation between particle D and β, quartiles for each particle D interval were calculated (Fig. 14). The particles were concentrated at |βR|<30∘ regardless of D, and median values were stable when D<1 mm for the entire particle shape types; however, fluctuation increased with D (Fig. 14a). The |σβ| values gradually increased from 10 to 13∘ when 0.3<D<1.3 mm, and variability was greatest around the center (13∘). This increase in |σβ| would not be expected in the case of a relatively small number of particles (Fig. 8a), since their standard deviation is largely maintained at about 13∘ regardless of data set size.

Variability in the median values for individual particles was more apparent. The values converged around 0∘ but fluctuation increased with greater D from the zero line. The median |βR| values exceeded 3, 5, 10, and 15∘ when D>1 mm, 1<D<2 mm, 2<D<3 mm, and D>3 mm, respectively (Fig. 14b).