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Volcanic plume parameters used in this paper. Table 1 4.1. Large Eddy Simulation results ells are utilized. The minimum vertical cell size is 6 = D/32, and a trading factor of 1.00187 is imposed. The azimuthal resolution is ‘onstant and equal to +7 (5.625 degrees). The resulting total num- yer of cells is 10,747,904. We also performed simulations with the same mesh topology but at lower resolution: mid-res (6=D/16) ind low-res (6=D/8).

Fig. 1. 3D numerical simulation of the WP case, 400 s after the beginning of the eruption (initial conditions as in Table 1). The two isosurfaces in 3D rendering correspond to the threshold €;= 107’ of the fine (white) and coarse (sand colored) ash volume fractions. The maximum height of the fine class isosurface is approximatively 12.1 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis. See color palette for volume concentration values.

Fig. 2. Vertical section of the instantaneous value of the mixture velocity modulus (in logarithmic scale) at t=400s and velocity streamlines. Fig. 3a shows the distribution of C; for the coarsest particles at t=400s. The color scale is logarithmic and symmetric with respect to 1, which corresponds to the zero preferential concentration. For C;<1, the mixture is relatively depleted of particles (green to blue scale); for Cj>1, particles are clustered (green to red scale). The mass fraction of the coarsest particles is up to 5 times larger (red zones) and 20 times smaller (blue zones) than the value it would have in absence of preferential concentration (green zones). This behavior affects the mixing and entrainment process, as it will be demonstrated in the following sections, where the mean plume Analysis of the influence of the averaging time-window (see, e.g. Esposti Ongaro and Cerminara, 2016) has shown that the averaging window size is much more important than the initial averaging time to, because long period fluctuations must be smoothed out. Therefore, we have taken the largest possible time window, even though a time- window [600-660] s was recommended and used in the inter- comparison study. On the other hand, t>) = 300 s is enough for the WP to reach its maximum height. The updraft is stationary beyond this time, even though the umbrella is not fully developed. However, this aspect of the analysis deserves more attention in the future.

Fig. 3. Distribution of Ccoarse (a), ANd Cfine (b), for the coarser and finer particles across a vertical section at t= 400 s (cf. Eq. (20)).

(b) Fine particles: preferential concentration Cane

Fig. 4. (First part) profiles of the integral variables of WP. Here we compare them by changing the resolution from 6= D/32 (high-res) to 6=D/16 (mid-res) and 6=D/8 (low-res). In the low-res case, we also present the results obtained with the [dusty] model. All runs adopt the [dynWale] SGS model.

M. Cerminara et al. / Journal of Volcanology and Geothermal Research xxx (2016) xxx-Xxx Fig. 5. Entrainment coefficient k for the WP case in the plume region 2+6 km above the inlet. This is a zoom of Fig. 4f . The points represent the results from the high-res simulation, and the best fit to this data. Fit results for the other simulations performed are also shown.

M. Cerminara et al. / Journal of Volcanology and Geothermal Research xxx (2016) xxx-Xxx Fig. 6. Behavior of conserved quantities of ASHOD as calculated from 3D simulations of the WP scenario. Here we modified some of the plume parameters to obtain bette: behavior of the conserved quantities (see discussion): dg= 1.65 > q= 4.62, Yc= 0.252 > yc= 0.292. Moreover, we used the value of vy, = 2.66*10~* obtained averaging the atmospheric profile: Vo=1.40 « 10 * Hz. Each point corresponds to a different height, evolving as indicated by the arrow: from the vent to the maximum height, through the NBL

Fig. 7. 3D numerical simulation of the SP case, 1000 s after the beginning of the eruption (inlet conditions are in Table 1). The two isosurfaces in 3D rendering correspond to the threshold €;= 107’ of the fine (white) and coarse (sand colored) ash volume fractions. The maximum height of the fine class isosurface is approximatively 43 km. The two-dimensional plots represent the distribution of the volume concentration of coarse (left) and fine (right) particles across vertical orthogonal slices crossing the plume axis. See color palette for volume concentration values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Vertical section of the instantaneous value of the mixture velocity modulus (in logarithmic scale, [m/s]) at t= 1000 s from the beginning of the eruption, and corresponding veloci streamlines. The most evident structure, specific to the SP, is the large toroidal eddy under the umbrella cloud. Less apparent, but still very important, is another toroidal eddy developing around the jet, under the suspended flow. These two features are present in the instantaneous fields as pulsat- ing turbulent structures, however they are also clearly recognizable in the time-averaged fields (e.g. Dobran et al., 1993; Cerminara, 2016). Fig. 8 shows the streamlines of the velocity field in a vertical slice of the whole simulated plume. Re-entrainment plays an important role, causing some of the settled and collapsing particles to be re-entrained into the plume and be transported upwards again (Dobran et al., 1993). Velocity fluctuations are stronger along the plume margins, while the velocity appears to be more stable near the plume axis. The jet region exhibits a complex behavior, being surrounded by a collapsing layer, a re- entrainment region and an ascending co-ignimbrite (Cerminara, 2016). A quantitative discussion of the influence of the plume collapse and par- ticle re-entrainment can be found in the following sections, where we

Fig. 9. Distribution of Ccoarse (a), aNd Cyne (b), for the coarser and finer particles across a vertical section at t= 1000 s (cf. Eq. (20)).

(b) Fine phase preferential concentration

Fig. 11. Infrasonic spectrum of the pressure fluctuations as measured from a probe placed at the ground level 15 km from the vent center, in the time window t€[0,13]min. The frequency is expressed both in its dimensional (f) and non-dimensional (Str =f D/U) formulation. Pressure fluctuations E, are represented both in logarithmic scale and in dB scale relative to 20 uPa. The slope Str~''/* is shown by the bold dashed line. The experimental large-scale similarity spectrum is presented by the bold solid line (see Tam et al., 1996). The inset shows the pressure fluctuation signal in kPa until t= 13min. t€[0,13]min. As for real infrasound signals (Matoza et al., 2009), the pressure signal is convoluted with a kernel window cutting the frequen- cies smaller than 0.03 Hz, to study the pressure fluctuations only. The resulting spectrum is compared with the similarity spectrum generated by large-scale turbulence measured by Tam et al. (1996). The fit is in satisfactory agreement and comparable with results obtained from di- rect volcanic observation (see Matoza et al., 2009). We obtained a peak in E,(Strp), for Strp = 0.32 + 0.01, a value comparable with that ex- pected from experimental results (see Lesieur et al., 2005; Cerminara et al., 2016). We also show the slope Str~ !!/4 measured by Tam et al. (1996) for supersonic jets for Str>2.5 Strp. It is worth noting that our

Fig. 10. Infrasonic signal generated by the turbulent eddies of the SP. Here the infrasonic perturbation is visualized using the magnitude of the acceleration field ag, in logarithmic color scale [m/s?], 120 s after the eruption started. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. (First part) Influence of the mesh resolution on the profiles of the integral variables of SP. Here we compare them by changing the resolution from 6 = D/32 (high-res) to 6=D/1¢ (mid-res) and 6=D/8 (low-res). All runs adopt the [dynWale] model.

Fig. 13. (First part) Influence of the subgrid LES model on the profiles of the integral variables of SP. Here, we compare them using high and mid resolutions. In the high resolution case, we present the results obtained without the subgrid model [noM], while in the mid resolution case [moin] and [oneEqEddy] are shown. M. Cerminara et al. / Journal of Volcanology and Geothermal Research xxx (2016) xxx-Xxx

M. Cerminara et al. / Journal of Volcanology and Geothermal Research xxx (2016) xxx-Xxx There are a few differences between the results shown in Fig. C.18: model and the partial collapse dynamics in the jet region. In the 3D sim- ulation we have found an entrainment coefficient larger than that used in the integral model. This means that the integral plume model is dilut- ing more slowly than in 3D simulations, and therefore the temperature remains higher, the velocity is larger and the plume goes higher, reaching a NBL 36% higher than that provided by 3D simulations. As in the WP case, assuming a constant entrainment coefficient above the NBL leads to incorrect behavior, where the plume radius diverges in- stead of going to zero. This discrepancy adds another source of error to the plume description, which in 3D simulations does the opposite, leading to a decrease in the plume height.

Fig. 14. (First part) Influence of the kinematic decoupling model on the profiles of the integral variables of SP. Here we compare results obtained using the [eqEu] model with those from the [dusty] model. The former is reported in the high and mid resolution case, whereas the latter in the mid resolution case.

et al., 2016) the ability to accurately reproduce the turbulent spectrum in both Direct and Large-Eddy Simulation. In particular, the model is able to reproduce not only the averaged dynamics of a laboratory forced plume but also the statistics of turbulent fluctuations, controlling the mixing rate. Such results are encouraging and strongly support the reliability of the adopted numerical model for volcanic plume simulation. However, Fig. 15. Entrainment coefficient k of SP below the NBL: 0+ 16 km. This is a zoom of Fig. 12f. We show the fit graph obtained at high resolution with the [dynWale] subgrid model, and report in the legend the fit result for the other simulations performed.

it is worth recalling that volcanic plumes are characterized by a Reynolds number several orders of magnitude larger than a laboratory plume and that atmospheric stratification strongly characterizes volcanic plume dy- namics, making an exact scaling of laboratory and DNS experiments ex- tremely difficult. Despite these limitations, we have demonstrated in this work (Figs. 10 and 11) that high-resolution numerical simulations are able to reproduce some qualitative and quantitative features of the in- frasonic spectrum emitted by volcanic plumes, an observable strictly linked to the (compressible) turbulent regime.

M. Cerminara et al. / Journal of Volcanology and Geothermal Research xxx (2016) xxx-Xxx Fig. 16. Behavior of conserved quantities of the integral model as calculated from 3D simulations of the SP scenario. Here we modified some of the non-dimensional plume parameters introduced in Appendix C.2 to obtain better behavior of the conserved quantities (see discussion): dg=0.473 > dg=2.7, y= 0.345 > y- = 0.445. Moreover, we used the value of vy 9 =3.0*10~? obtained averaging the atmospheric profile: N=2.61 « 107? Hz. The arrows indicate the vertical upward direction of the flow in the plume from the vent to the NBL.

Figure 17 – uploaded by Matteo Cerminara

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