1.1 Drag coefficient of a spherical particle as a function of Reynolds number [8]. . . 15 1.2 Drag coefficient for spherical and nonspherical particles
trans-formed in generalized drag coefficients and generalized Reynolds number [11]. . . 18 1.3 (a) Staggered grid adopted by the PDAC code used to describe
the carrier flow field. Three different interpolations of the carrier flow field are required to determine the local radial velocity (b), the vertical velocity (c), and the pressure gradient (d) of a particle with position (x; z) [9]. . . 23 1.4 Sketches of a strong and a weak plume [7]. . . 25 1.5 Plume of Mt.Etna eruption of 24 November 2006. . . 26 1.6 Cloud ash of 24 November 2006 eruption captured by MOSIS
satellite, source NASA . . . 26 1.7 KHI present in natural phenomena. . . 27 1.8 Mechanism of formation of KHI. . . 27 1.9 Setting table of WRF for the generation of the carrier flow
field [18]. . . 28 1.10 Wind field at the beginning of the Lagrangian simulation. The
Kelvin-Helmholtz instability are found to originate in corre-spondence of the volcanic plume, centered at ∼ 4000m a.s.l. . 29 1.11 Wind field after little more than an hour of simulation. The
KHI are evident and they will remain stable also in the next times. . . 30
1.12 Wind field at time t = 9100s. Not significant differences can
be found with respect to the previous instant of time. . . 31
1.13 Wind field at the end of the simulation, KHI is still present. . 32
1.14 Trajectories of two groups of particles of different diameters. . 33
2.1 Clenshaw-Curtis grid used in the present work [1]. . . 43
2.2 Plot of f (x1, x2, x3) = x 1/2 1 x3+ 3x33+ 10x2 evaluated for 1000 random samples. . . 45
3.1 Common instruments used to evaluate the grain size of a deposit. 48 3.2 Udden-Wentworth grain-size classification of terrigenous sedi-ments [22]. . . 49
3.3 Total grain-size distribution of tephra-fallout deposits esti-mated for the Plinian eruption of Mount St.Helens1980 [7]. The scale reported in abscissas is the Krumbein scale. . . 50
3.4 Discrete number frequency distribution. . . 51
3.5 Number of parcels distribution of a sediment composed by 10000 parcels with a mean diameter of 0φ and a Standard deviation of 1φ. . . 51
3.6 Density trend as a function of the particle diameter [7]. . . 52
3.7 Gaussian distribution characterized by three different value of skewness: equal to zero, positive and negative skewness. . . . 55
3.8 Variation of the Gaussian PDF according to the kurtosis value. 56 3.9 Representation of the computational domain. . . 57
3.10 Positions from which the parcels are released. Each red point represents an initial position of the parcels. . . 58
3.11 Different sketches of parcels distribution. . . 61
3.12 Grain size distributions referred to Mount Etna eruptions. . . 62
3.13 Particle sphericity versus roundness [14]. . . 64
3.14 Input distribution variables . . . 65
4.1 CDFs for 5 different quadrature orders. . . 69
4.3 Image of a polynomial response function present in the output file of DAKOTA. For each coefficient, the Pi are the
unidimen-sional Legendre polynomials which are mixed together in order to generate a multivariate basis. . . 73 5.1 Plot of the parcels positions at time t=7600s. The input
parameters of this simulation are the mean ones: µ = 1φ, σ = 1.5φ, ψ = 0.7 . . . 75 5.2 Parcels positions at time 9100 s. . . 75 5.3 Parcels positions at time 10600 s. . . 75 5.4 Parcels size distribution plotted at three time instants. The
simulation taken into account is still the one with the mean input values. . . 76 5.5 Parcels size distribution in each region of the domain
associ-ated with the input value of σ and ψ equal to 1.5φ and 0.7 respectively, while µ varies from the minimum value of 0φ to the maximum one of 2φ. Also the mean solution associated to µ = 1φ is reported. . . 78 5.6 Parcels size distribution associated to µ = 1φ and ψ = 0.7
(mean input value), while σ varies from its minimum to its maximum value, which are 1.3ψ and 1.7ψ respectively. . . 81 5.7 Parcels size distribution in each region of the domain
associ-ated with the input value of µ and σ equal to 1φ and 1.5φ respectively, while ψ varies from the minimum value of 0.5 to the maximum one of 0.9. . . 83 5.8 Sketch of the polynomial response function associated to the
mean value of the parcels diameter in Stripe 1-Air. The do-main reported on x-y axis refers to the range of existence of Legendre polynomials[-1,+1]. . . 87 5.9 Mean parcels diameter associated to Stripe 1-Ground. The
domain reported on x-y axis refers to the range of existence of Legendre polynomials[-1,+1]. . . 88 5.10 CDFs (a) and PDFs (b) related to the mean parcels diameter. 90
5.11 CDFs (a) and PDFs (b) of the standard deviation of the parcels distribution. . . 93 5.12 Scketchs of the CDFs (a) and PDFs (b) associated with the
skewness value in each cell. . . 95 5.13 CDFs (a) and PDFs (b) which show the trend of the kurtosis
value on the domain. . . 97 5.14 CDFs (a) and PDFs (b) of the number of parcels distribution
over the domain. . . 99 5.15 Main Sobol Indices of the output values concerning the air
distributions.The variability of the output values (i.e. mean diameter of parcels distribution µ, standard deviation σ, skew-ness (Sk), kurtosis (Ku) and number of parcels (P)) is due to the variation of the input values of µ and ψ. . . 102 5.16 Main Sobol Indices of the output values concerning the ash
deposit on the ground. In this case the contribution of ψ variation is leader also with respect to µ variation. . . 102 5.17 Total Sobol Indices related to the air parcels distributions. . . 103 5.18 Total Sobol Indices related to the air parcels distributions. . . 103 5.19 Main Sobol Indices and Total Sobol Indices computed for the
output values of Stripe 1-Air. The three graphs represent the contribution to the output variability due to µ, σ and ψ re-spectively . . . 104 5.20 Comparison between Main Sobol Indices and Total Sobol
In-dices computed for the output values of Stripe 2-Air. . . 105 5.21 Comparison between Main Sobol Indices and Total Sobol
In-dices computed for the output values of Stripe 3-Air. . . 105 5.22 Comparison between Main Sobol Indices and Total Sobol
In-dices computed for the output values of Stripe 4-Air. . . 106 5.23 Comparison between Main Sobol Indices and Total Sobol
In-dices computed for the output values of Stripe 1-Ground. . . . 106 5.24 Comparison between Main Sobol Indices and Total Sobol
5.25 Comparison between Main Sobol Indices and Total Sobol In-dices computed for the output values of Stripe 3-Ground. . . . 107 5.26 Comparison between Main Sobol Indices and Total Sobol
In-dices computed for the output values of Stripe 4-Ground. . . . 108 5.27 Trend of CD as a function of the particles diameter for the
three different values of sphericity. . . 109 5.28 Trend of CD as a function of the particles diameter varying
the mean input diameter. . . 109 5.29 Drag coefficient as a function of Reynolds number. . . 110
2.1 Askey Scheme [1]. . . 38 3.1 Classification of grain size distribution considering the
stan-dard deviation [16]. . . 54 3.2 Classification of grain size distribution as a function of the
skewness value [16]. . . 55 3.3 Summary of the input parameters together with their range
of variation. . . 65 3.4 Summary of the output quantities of interest computed in each
cell. . . 66 4.1 Total number of quadrature points associated to each
quadra-ture order. . . 68 4.2 Variation of the maximum polynomial order and of the
re-sponse coefficients number as a function of the quadrature order. . . 69 4.3 Variation of the mean value of parcels diameter as a function
of the quadrature order. . . 70 4.4 Simulation time for the different quadrature order. . . 70 4.5 Weights values associated to the 7 quadrature points. . . 71 5.1 Maximum and minimum values of the initial distributions. . . 77 5.2 Statistical parameters of parcels distribution referred to the
5.3 Statistical parameters of parcels distribution referred to the mean input value of µ (1φ). . . 79 5.4 Statistical parameters of parcels distribution referring to the
upper input value of µ (2φ). . . 80 5.5 Output results referred to the input triplet: µ = 1φ, σ = 1.3φ
and ψ = 0.7. . . 81 5.6 Output results referred to the input triplet: µ = 1φ, σ = 1.5φ
and ψ = 0.7. . . 82 5.7 Output results referred to the input triplet: µ = 1φ, σ = 1.7φ
and ψ = 0.7. . . 82 5.8 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ
and ψ = 0.5. . . 84 5.9 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ
and ψ = 0.7. . . 84 5.10 Output values referred to the input triplet: µ = 1φ, σ = 1.5φ
and ψ = 0.9. . . 85 5.11 Mean value and standard deviation referred to the PDFs
as-sociated to the mean parcels diameter in every cell. . . 91 5.12 Mean value and standard deviation referred to the PDFs
as-sociated to the standard deviation of the parcels distribution in every cell. . . 94 5.13 Mean value and standard deviation referred to the PDFs
as-sociated to the skewness value of the parcels distribution in every cell. . . 96 5.14 Mean value and standard deviation referred to the PDFs
asso-ciated to the kurtosis value of the parcels distribution in every cell. . . 98 5.15 Mean value and standard deviation referred to the PDFs
as-sociated to the number of parcels present in every cell. . . 100 5.16 Values of the relative mass fractions computed in every cell. It
