Simulation Set Up
In this chapter we present the choices we have made in order to develop a methodology suitable for the study of ash dispersion. First of all, we introduce the tool used to describe the mass partitioning of the ash particles, the so-called Grain Size Distribution. Then we present the way in which the computational domain has been defined and treated with the aim to have a clear view of what happened to the particles. Finally, we show the input and the output parameters we have chosen as object of our analysis.
3.1
Grain Size Analysis
When we deal with the texture of a volcanic ash deposit, a variety of char-acteristics are involved. However the grain size is the most fundamental property of sediment particles, affecting their entrainment, transport and deposition. Grain Size Analysis gathers all the techniques used in the study of an ash deposit.
The importance of the calculation of the grain size distribution for a tephra fall deposit lies in the following motivations:
• possibility to infer the eruption style by studying the connection be-tween the particle size and the initial gas content and water magma interaction processes;
• since the grain size curves are a consequence of the sedimentation, it is possible to use the size distribution curves to investigate the dynamics of the plume;
• the total grain size distribution is a crucial element in the quantifica-tion of the hazard to the populaquantifica-tions and in the development of safety plans related to eruptive phenomenons. It is also fundamental in order to investigate the level of particulate pollution dangerous for human health.
The determination of total grain-size distribution is a very complex task; several difficulties can raise from the scarcity of samples, from the big variabil-ity of data concerning a particular area, from the lack of an unique method-ology of calculation.
Also is very difficult to reconstruct a complete grain size distribution because of the problems concerning the sampling of particles below 63µm diameter.
The various techniques employed in grain size determination include: di-rect measurement, dry and wet sieving, measurement by laser granulome-ter, X-ray sedigraph and Coulter counter (some instruments are reported in Fig.(3.1)).
(a) A set of Sieves of different sizes. (b) Laser granulometer
(c) X-ray sedigraph (d) Coulter counter
In the problems concerning the grain size analysis, arithmetic grain scales are rarely used since too much emphasis is placed on coarse sediment and too little on fine particles [16]. This problem can be solved using a logarithmic (geometric) scale which allows to place equal emphasis on small differences in fine particles and larger differences in coarse particles. The scale most frequently used is the so-called Krumbein scale, proposed in 1922 by William Wentworth [22] and enhanced in 1934 by Krumbein [13].
According to Krumbein, the diameter of a particle can be expressed as:
φ = − log2(D D0
), (3.1)
where D is the grain diameter in millimeters and D0 is a reference
diam-eter equal to 1 mm.
In this scale, 1 millimeter corresponds to 0 φ. Sizes greater than 1mm have negative φ values while those less than 1mm have positive φ values.
An example of classification of grains according to the different scales is reported in the Fig.(3.2).
Figure 3.2: Udden-Wentworth grain-size classification of terrigenous sedi-ments [22].
When dealing with volcanic ash particles, it is important to consider that similar amount of mass can be present in the atmosphere for particles in a large range of diameters in the Krumbein scale. This means that if the range of ash diameters covers a scale from 10−6m to 10−3m, the number of particles with a diameter equal to 10−6m necessary to balance the mass of one particle of 10−3m diameter is 109 units. Such number of particles results
to be unmanageable by any Lagrangian code. For this reason, we have not considered individual ash particles but the objects of our study are pack-ages of particles (parcels) and all the parcels are assumed to have the same mass. In particular each parcel is composed by an indeterminate number of particles having the same diameter, so each parcel is representative of a particular diameter. This stratagem becomes necessary when we deal with a huge number of particles of very different dimensions.
3.1.1
Particles Size Distribution
Various representations can be employed in order to study the particles size of an ash deposit. The tool most commonly used by scientists is the Grain Size Distribution. After the sediment sample has been divided into a number of size fraction ∆φ (the mid-point is considered the representative diameter of each interval), the size distribution is constructed evaluating the weight or the volume percentage of sediment in each size fraction (see Fig.(3.3)).
Figure 3.3: Total grain-size distribution of tephra-fallout deposits estimated for the Plinian eruption of Mount St.Helens1980 [7]. The scale reported in abscissas is the Krumbein scale.
Another approach is the frequency distribution of the diameters, which is actually a PDF of the distribution. This graph is built dividing the number of particles counted in every diameters interval for the total number of the sediments (see Fig.(3.4)). In order to compare different sediments, grain size distributions are most frequently performed assuming an ideal Gaussian distribution for the mass fraction.
−60 −4 −2 0 2 4 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 φ Frequency Distribution
Figure 3.4: Discrete number frequency distribution.
In our work we have chosen to study the dispersion of the particles con-sidering the number of the parcels present in a specific region of the domain at a certain time. So, being known the sample we want to analyze, each parcel is positioned in its size interval ∆φ. The result of this operation is the number of parcels distribution, which is a kind of grain size distribution.
−50 −4 −3 −2 −1 0 1 2 3 4 50 100 150 200 250 300 350 φ
Number of parcels distribution
Figure 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.
The settling of particles is controlled not only by their size, but also by their density. The density of fine particles is difficult to be measured and a complete data set is not available. The pumice particles1, the only taken into account in the present work, are highly vesicular fragments, so their density varies significantly with the size. Smaller fragments, which are less vesicular, are more dense than the coarse one. Decreasing the diameter up to particles of 7φ diameter, the density increases linearly, and, for the finer fragments, remains constant and equal to the lithics one (see Fig.(3.6) [7].
Figure 3.6: Density trend as a function of the particle diameter [7].
1Pumice is a volcanic rock of the class of juvenile rocks. It is generated by the magma
during the eruption, while the lithics are fragments derived by the consolidated magma of previous eruptions.
3.1.2
Statistical Parameters
The study of a sediment deposit is usually made in a statistical way. The statistical parameters used are described in the next paragraphs.
Mode
The mode corresponds to the peak in the frequency distribution curve (if the curve has an arithmetic scale) and it represents the most frequently-occurring particle diameter. This parameter is not commonly use since its difficulty to be measured (no good mathematical formulas exist for its de-termination), it is also independent from the grain size of the rest of the sediment, therefore it is not a good measure of overall average size.
Median
The diameter corresponding to the 50% mark on the cumulative frequency curve, is called median. This means that half of the particles are coarser than the median, and half are finer. Even if it is the easiest value to determine, it is not able to reflect the overall size of the sediment, since the extremes values of the distribution are not considered. For this reason the median is not frequently used.
Mean value
The mean value (µ) is the statistical parameter that allows the best view of the overall particles size. If the total number of parcels (NP) located in a
region of the domain is known, having indicated with φi the diameter of the
i-th parcel, it is possible to compute the mean value as follows:
µ = PNP
i=0φi
NP
Standard Deviation
The sorting of the sediment from the average can be computed using the standard deviation (σ): σ = s PNP i=0φ2i NP − µ2. (3.3)
A classification of an ash deposit as a function of the Std. Dev. is re-ported in Tab.(3.1).
σ size range φ Description of the sorting
under 0.35 very well sorted
0.35-0.50 well sorted
0.50-0.71 moderately well sorted
0.71-1.00 moderately sorted
1.00-2.00 poorly sorted
2.00-4.00 very poorly sorted
over 4.00 extremely very poorly sorted
Table 3.1: Classification of grain size distribution considering the standard deviation [16].
Skewness
From a mathematical point of view, the skewness is defined as the ratio between the central moment of the third order and the third power of σ:
sk = PNP i=0φ 3 i NP − 3µ PNP i=0φ 2 i NP + 2µ3 σ3 . (3.4)
Graphically this parameter indicates the symmetry of the frequency dis-tribution curve. While for a symmetric disdis-tribution the value of the skewness is equal to zero, negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right, as sketched in Fig.3.7:
Figure 3.7: Gaussian distribution characterized by three different value of skewness: equal to zero, positive and negative skewness.
Concerning a sediment deposit, grain size distributions having excess fine material (a tail to the right according to the Krumbein scale) have positive skewness and those with excess coarse material (a tail to the left) have neg-ative skewness. A classification of an ash deposit through its skewness is reported in Tab.(3.2).
Skewness range Description of the sorting
1.00/0.30 strongly fine skewed
0.30/0.10 fine skewed
0.10/-0.10 near symmetrical
-0.10/-0.30 coarse skewed
-0.30/-1.00 strongly coarse skewed
Table 3.2: Classification of grain size distribution as a function of the skew-ness value [16].
Kurtosis
Kurtosis is a measurement of the degree of peakedness of a distribution and it is defined as the ratio between the central moment of the fourth order and the fourth power of σ:
ku = PNP i=0φ 4 i NP − 4µ PNP i=0φ 3 i NP + 6µ2 PNP i=0φ 2 i NP − 3µ4 σ4 . (3.5)
In this work the kurtosis of a Gaussian distribution is set equal to zero (Mesokurtic), so data sets with positive kurtosis (Leptokutic) tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with negative kurtosis (Platykurtic) tend to have a flat top near the mean rather than a sharp peak, see Fig.(3.8).
Figure 3.8: Variation of the Gaussian PDF according to the kurtosis value. Kurtosis measures the ratio of the sorting in the extreme values of the dis-tribution compared with the sorting in the central part; thus, it is a sensitive and valuable test of the normality of the distribution. In fact distributions that seems to be normal by the values of their skewness, revel to be non normal when the kurtosis is computed.
3.2
Definition of the computational domain
The domain in which the particle dispersion is simulated has an horizontal extension of about 40 Km and a vertical one of about 6 Km. To better investigate the movement of the parcels, the domain has been divided into 4 vertical stripes and, in each stripe, a distinction between the air and the ground has been made, the ground is placed at the height of 0 Km, see Fig.(3.9). The inlet represents the point from which the parcels are released and it is located 2Km from the volcanic vain. As said before, we imagine to deal with a weak plume and the inlet is the section of the plume from which we start to observe the ash dispersion.
0 0.5 1 1.5 2 2.5 3 3.5 4 x 104 0 1000 2000 3000 4000 5000 6000 x
Height (z) Stripe 1 Stripe 2 Stripe 3 Stripe 4
GROUND INLET
From the beginning of the simulation, the parcels are released every 10 seconds in groups composed by 41 units. The parcels of each group are equally spread on 41 different heights in the range [3300m, 4300m] as repre-sented in Fig.(3.10): 99 99.5 100 100.5 101 3200 3400 3600 3800 4000 4200 4400 x Height (z)
Figure 3.10: Positions from which the parcels are released. Each red point represents an initial position of the parcels.
3.3
Definition of the Uncertain Input
Parame-ters
In this section the uncertainty input parameters are presented together with the range of their variation. What we want to investigate is how the at-mospheric ash dispersion changes with varying the initial conditions. These conditions are those LPAC need in order to evaluate the trajectory of each parcel. In particular what we set in the LPAC input file are:
• initial velocity components of the parcels;
• mean value of the initial grain size distribution (µ);
• standard deviation of the initial grain size distribution (σ); • density of the particles ;
• sphericity of the particles (ψ); • initial position of the parcels.
Since the dispersion and the deposition of the volcanic ash are mainly influenced by the grain size distribution, the uncertain input parameters we have chosen are: the mean value of the input distribution (µ), its standard deviation (σ) and the sphericity of the particles (ψ). The initial velocities have been set equal to zero, the density has been evaluated as a function of the parcels diameters (see Fig.(3.6)), while the initial positions are those specified in the previous paragraph.
3.3.1
Creation of the input parcels size distribution
First of all, it is necessary to create the input parcels size distribution associ-ated to the selected initial conditions. So, having chosen suitable values for µ and σ, the first step is the creation of the initial grain size distribution of the ash cloud. Since we have not considered the mass of the particles (all parcels have been assumed to have the same mass), the grain size distribution is actually linked to the size parcels distribution.
If, for example, the values chosen are: µ = 0φ, σ = 1φ the number of parcels distribution is built by the generation of a set of Np = 40000 random
diameters whose frequency distribution is actually the PDF characterized by the previously fixed values (see Fig.(3.11)). The value of the parcels diam-eter varies from −4φ to 4φ and, because of the symmetry of the Frequency Distribution around the mean value, it is possible to see that all the range of parcels dimensions has been correctly generated and discriminations have not been done with respect to the finer particles, those with the lower probability to be sampled.
−50 −4 −3 −2 −1 0 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 φ CDF
(a) Cumulative Distribution Function of the parcels that form the initial ash cloud. According to the Krumbein scale the coarse particles presents a negative value of φ and they have a higher probability to be sampled.
−50 −4 −3 −2 −1 0 1 2 3 4 5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 φ Frequency Distribution
(b) Frequency Distribution Function of the parcels that form the initial ash cloud. −50 −4 −3 −2 −1 0 1 2 3 4 5 500 1000 1500 φ
Number of parcels distribution
(c) Distribution of the parcels diameters. The Histogram is composed by 100 bars and it is possible to notice that the Gaussian distribution
3.3.2
Choice of the range of variation
Mean Value and Standard Deviation
Varying the initial conditions in term of µ, σ and ψ, different ash clouds are generated. The choice of the range of variation of the mean value and the standard deviation has been made having considered the information reported in literature about ash deposits of Mount Etna. In Fig.(3.12) are sketched the results of some works concerning this problem:
(a) Total grain-size distribution of the deposit between 27 October and 30 De-cember 2006; the statistic value are: mode 0φ and std 3φ, [3].
(b) The total grain-size distribution of the 4-5 September 2006 deposit, the statistic value are: mode 0.5φ and std 3.7φ, [2].
(c) Total grain-size distribution of the 21-24 July 2001 deposit, the statistic val-ues are: mode 2φ and std 1.9φ, [17].
Another distribution that can be considered is the one associated with the ash deposit sampled after the Mt. Etna eruption of the 2006 by Barsotti et all [5]. In this case the statistical values are: mode = 2φ and a std = 1.5φ.
Since the aim of this work is to expose the methodology which can be used to deal with this kind of problems, we have considered a quite large variation of the parameters. We have also assumed that the input distribution is a symmetric distribution, so the mode and the mean values are equal and the value of the skewness and kurtosis are supposed to be equal to zero. With the previous hypotheses, we have chosen a range of variation of the mean initial diameter from 0φ to 2φ, and a range of variation of the initial standard deviation from 1.3φ to 1.7φ
Even if these values are referred to the volcanic vain, using a plume model, we have observed that there is not a considerable change when we move to the point in which the parcels are released.
Sphericity
According to the formulation of the drag coefficient used in the present work, the sphericity is one of the main factor affecting the value of the CD. In
order to evaluate the effect of the shape factor on the dispersion, a reasonable range of ψ variation can be the one with a lower value of 0.5 and a upper value of 0.9. The consequences of this choice on particle shape are shown in Fig.(3.13). The two particle shape descriptor most commonly used are the sphericity and the roundness; in our work we have not specified the particles roundness and only the effects of the sphericity on the movement of ash particles have been evaluated.
The input parameters used in the UQ study are summarized in Tab.(3.3):
Input parameter Lower Bound Upper Bound
µ [φ] 0 2
σ [φ] 1.3 1.7
ψ 0.5 0.9
Table 3.3: Summary of the input parameters together with their range of variation.
Because of the lack of information, the PDF used to characterize the input variability is a uniform distribution as reported in the Fig.(3.14).
0 1 2 0 1 2 3 µ Input PDF
(a) Uniform input PDF of the input variable µ 0 1 2 0 1 2 3 σ Input PDF
(b) Uniform input PDF of the input variable σ 0 1 2 0 1 2 3 ψ Input PDF
(c) Uniform input PDF of the input variable ψ
3.4
Output quantities of interest
The numerical code LPAC has been set in order to give in output the pa-rameters that allow us to reconstruct the dynamics and the deposit of the particles. In particular, for a specific region of the domain, the information we obtain from LPAC are:
• The number of the parcels;
• The mean value of the parcels diameters;
• The standard deviation of the parcels diameters; • The skewness of the parcels diameters;
• The kurtosis of the parcels diameters.
Also, referring to the domain previously specified, these outputs have been computed in every stripe and, for each stripe, a distinction between air and ground has been made. A summary is reported in Tab.(3.4), where it is possible to see all the output of the code.
Stripe 1-Air Stripe 2-Air Stripe 3-Air Stripe 4-Air
Mean (µ) Mean (µ) Mean (µ) Mean (µ)
Std-Dev (σ) Std-Dev (σ) Std-Dev (σ) Std-Dev (σ)
Skewness Skewness Skewness Skewness
Kurtosis Kurtosis Kurtosis Kurtosis
Number of parcels Number of parcels Number of parcels Number of parcels
Stripe 1-Ground Stripe 2-Ground Stripe 3-Ground Stripe 4-Ground
Mean Mean Mean Mean
Std-Dev Std-Dev Std-Dev Std-Dev
Skewness Skewness Skewness Skewness
Kurtosis Kurtosis Kurtosis Kurtosis
Number of parcels Number of parcels Number of parcels Number of parcels Table 3.4: Summary of the output quantities of interest computed in each
cell.
The results have been analyzed at three different time instant: 7600s, 9100s, 10600s from the beginning of the simulation.
