Introduction
In order to investigate the problem of Uncertainty Quantification (UQ) in volcanic ash dispersion and deposition, Chapter 1 of this thesis presents the Lagrangian code, LPAC, used to perform the 2D simulations. The trajec- tories of the ash particles have been estimated by numerically solving the Basset-Boussinesq-Ossen Equation of motion [8], in which we have taken into account the drag force, the pressure gradient, the gravity and the Von- Mises forces. In particular the drag coefficient of ash fragments has been shaped as a function of the Reynolds number and of the particles sphericity [11].
The wind field used to perform the simulation has been generated by WRF code and it tries to reproduce the atmospheric conditions related to the Mount Etna eruption of 24 November 2006. In this occasion Kelvin- Helmholtz instabilities have been observed at the height of about 4000m from the sea level [18].
Chapter 2 presents the mathematical tool used in the Uncertainty Quan- tification process, the Generalized Polynomial Chaos Expansion method (PCEg) [1]. Having characterized the uncertainty parameters by an appropriate prob- ability density function, this stochastic expansion technique is able to build the output quantities of interest as polynomial functions which reflect the propagation of the input uncertainty in the model. The UQ has been carried out by DAKOTA toolkit, a code developed by Sandia Lab. that permits to conduct also the Sensitivity Analysis (SA) of the response functions.
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Introduction 11
Having to treat the phenomenon of ash dispersion and deposition, in Chapter 3 the common tools used by sedimentologists to study the charac- teristic of a tephra fall deposit are presented. Since the particles dimension is the most important factor that influences the ash movement, the Total Grain Size Distribution curves are briefly described together with the Fre- quency Distribution ones and the Number of Particles Distributions [10].
These graphs are used in order to organize the fragments distribution ac- cording to their size expressed using the Krumbein scale. The statistical parameters that characterized the particles distributions (i.e. the mean di- ameter, its standard deviation, skewness and kurtosis) are then introduced and their physical interpretation is shown.
It is fundamental to specify that, in the numerical simulation, we do not consider the single particles, but we study the dispersion of ash packages (parcels). Each parcel is composed by an indeterminate number of particles so that all the parcels have the same mass. This has been done in order to do not treat with a huge number of ash particles. Chapter 3 also presents the description of the domain used for the simulation. The total dimensions of the two-dimensional grid over which the particles dispersion take place are: 40Km horizontally and 6Km vertically. This space has been equally divided into 4 vertical equal stripes and the features and the distributions of the particles in air and of those deposted on the ground have been analyzed separately.
The parcels are released every 10 seconds in groups of 41 units equally spread from a height of 3300m to an height of 4300m. The simulation lasts 2 hours and the output quantities of interest are computed in every cell at three different time instants; however the results have been studied only in corre- spondence of the last time (once the Kelvin-Helmholtz instability has been generated, the output statistics are approximately steady).
The uncertain input parameters we have chosen to study are those related to the geometrical characteristics of the parcels input distribution: the mean diameter (µ), its standard deviation (σ) and the sphericity of the particles
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(ψ). From literature studies of tephra fall deposit of Mount Etna 2006 erup- tion [7, 6, 4, 3], the range of uncertainty for each parameter has been set to be the following way: µ = [0φ, 2φ], σ = [1.3φ, 1.7φ] and ψ = [0.5, 0.9]. Be- cause of lack of available information, the probability density functions used to describe the input parameters were chosen to be uniform in the ranges previously specified. In order to study the ash dispersion, the output quan- tities are the statistical parameters that permits to reconstruct the parcels distribution in every cell (i.e. the mean diameter, its standard deviation, skewness and kurtosis) and the number of parcels present in each cell.
Chapter 4 presents the setting procedure of DAKOTA. Since the input variables have been described by uniform PDF, the polynomials used by DAKOTA in order to build the response functions are Legendre polynomi- als. The low number of input variables allowed us to evaluate the polynomial coefficients using a Gaussian quadrature technique with 7 quadrature points.
The results of the simulation are illustrated in Chapter 5. First of all some deterministic solutions obtained by keeping constant an input value and varying the other two are presented. Then the results of the UQ analy- sis are considered. In particular the statistical parameters associated to the output values are plotted, together with their Cumulative Distribution Func- tions and their Probability Distribution Functions. Analyzing these data it is possible to observe that the initial variability of the input parameters tends to disappear with increasing distance from the inlet and the particles tend to have more uniform geometrical characteristics. Studying the results of SA (Main and Total Sobol Indices) the main parameters which affect ash dispersion and deposition are µ and ψ, while the initial sorting of the distri- bution σ does not play a fundamental role. While ψ uncertainty dominates the variation of the drag coefficient, µ characterizes the way in which the particles are coupled with the carrier flow. So, while larger sediments show a behavior which is almost independent of the wind field, the finer ones are more sensitive to the flow and thus they are able to reach the largest distance from the vain.
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