In this paper a new integral mathematical model for volcanic plumes, named
PLUME-MoM, is presented. The model describes the steady-state dynamics of a
plume in a 3-D coordinate system, accounting for continuous variability in
particle size distribution of the pyroclastic mixture ejected at the vent.
Volcanic plumes are composed of pyroclastic particles of many different sizes
ranging from a few microns up to several centimeters and more. A proper
description of such a multi-particle nature is crucial when quantifying
changes in grain-size distribution along the plume and, therefore, for better
characterization of source conditions of ash dispersal models. The new model
is based on the method of moments, which allows for a description of the
pyroclastic mixture dynamics not only in the spatial domain but also in the
space of parameters of the continuous size distribution of the particles.
This is achieved by formulation of fundamental transport equations for the
multi-particle mixture with respect to the different moments of the grain-size
distribution. Different formulations, in terms of the distribution of the
particle number, as well as of the mass distribution expressed in terms of
the Krumbein log scale, are also derived. Comparison between the new
moments-based formulation and the classical approach, based on the
discretization of the mixture in

In the past decades, numerical simulation of volcanic eruptions has greatly
advanced and models are now often able to deal with the multi-phase nature of
volcanic flows. This is the case, for example, with models describing the
dynamics of pyroclastic particles in a volcanic plume, or that of bubbles and
crystals dispersed in the magma rising in a volcanic conduit. Despite this,
in numerical models, the polydispersity associated with the multi-phase nature
of volcanic flows is often ignored or largely simplified

A theoretical framework and the corresponding computational models, namely,
the method of moments for disperse multi-phase flows, have been developed in
the past decades, mostly in the chemical engineering community

In this work we present an extension of the Eulerian steady-state volcanic
plume model presented in

This paper is organized as follows: in Sect. 2 we present the method of moments applied to two different descriptions of particle distribution. In Sect. 3 the equations of the model for the two formulations are described. Section 4 is devoted to the numerical discretization of the model and the numerical implementation of the method of moments. Section 5 presents the application of the model to three test cases with a comparison of the model results for different formulations of the plume model, and finally an uncertainty quantification and a sensitivity analysis are applied to model results.

In contrast to previous works, where the solid particles are partitioned in a
finite number of classes with different sizes

First, we present the method of moments for a particle size distribution

Given a particle size distribution

The particular definition of

We also note that the central moments (i.e., those taken about the mean) can
be expressed as function of the raw moments (i.e., those taken about zero as
in Eq.

The motivation for the introduction of the moments is to minimize
computational costs by avoiding the discretization of the size distribution
in several classes, and nevertheless to capture the polydispersity of the
flow through the correct description of the evolution of the moments

In the plume model, several quantities characteristic of the particles, such
as settling velocity, density and specific heat capacity, are also defined as
functions of the particle diameter, and thus we can define their moments as was carried out
for the particle size distribution

As a first example, we consider here the moments of particle density

The moments defined by Eq. (

In contrast to the approach used in

As carried out for particle density, it is possible to evaluate the
moments

Finally, it is possible to define the moments

While in chemical engineering, where the method of moments is commonly used,
the particle number distribution

In this case, the distribution

Again, it is possible to define the moments of other quantities

In this section we describe the assumption and the equations of the model. As
in

The equation set for the plume rise model is solved in a 3-D coordinate
system

Schematic representation of the Eulerian plume model. The dashed
black line represent the axis of the curvilinear coordinate

Following

Now, multiplying both the sides of Eq. (

From Eq. (

Now, following the same procedure, we reformulate the other conservation equations describing the steady-state ascent of the plume in terms of the moments of the continuous distributions of sizes, densities and settling velocities instead of the averages over a finite number of classes of particles with different size.

First of all, we derive the conservation equation for the mixture mass. As in
the plume theory, we assume that the entrainment, due to both turbulence in
the rising buoyant jet and to the crosswind field, is parameterized through
the use of two entrainment coefficients,

From Newton's second law and the variation of mass flux, we can derive also
the horizontal and vertical components of the momentum balance solved by the
model as

Now, following the notation adopted above and denoting with

From this expression, if we multiply all the terms at the numerator and the
denominator of the right-hand side by

Similarly, a gas constant

Finally, as in

Similarly to the distribution of particle number

In this case, the conservation of mass flux of particles with size

From the variation of mass flux, as was carried out for the distribution of particle
number

The formulation of the equations for the gas constant

The plume rise equations are solved with a predictor–corrector Heun's scheme

We observe that to calculate the right-hand side for both the predictor and
corrector step we need not only the moments

Here

The Wheeler algorithm, as presented in

A strategy that might overcome the problem of moment corruption (i.e., the
transformation during the integration of the moment-transport equations of a
realizable set of moments into an unrealizable one) is replacing unrealizable
moment sets as soon as they appear. An algorithm of this kind was developed
by McGraw

Thus, in both the predictor and corrector step, the following algorithm is
used:

The nodes

The quadrature formula (Eq.

The right-hand side of the ODE's system (Eq.

The solution is advanced with the predictor (or the corrector) step of the Heun's scheme.

For each particle family

We observe that if the

Initial conditions at the vent include the initial plume radius (

For the application presented in this work, the initial distribution

Given the parameters

We observe that if we introduce the following re-scaled variables for the
diameter, the mean and the variance:

From the expressions of the moments it follows also that, if the mass
concentration expressed as a function of the Krumbein scale has a normal
distribution, the Sauter mean diameter

Processes involving the mutual interaction between particles and the
interaction between the particles and the carrier fluid (friction and
cohesion between the particles; viscous drag; chemical reactions between
fluid and solid components) operate at the surface of the particles. For this
reason the Sauter mean diameter, based on the specific area of the particles,
is a convenient descriptor and it is important to remark that it differs from
the mean

When the Sauter mean diameter is used, also the variance and the standard
deviation SD should be based on the specific surface area

Finally, we note that if the particle density is constant and the mass
concentration expressed as a function of the Krumbein scale has a lognormal
distribution and both the Sauter mean diameter

Once the re-scaled mean and variance are known, we can obtain

When the initial distribution is expressed for the mass fractions instead of
the particle number, and the mass fraction written as a function of the
Krumbein scale has a normal distribution with mean

In this case, the moments

Input parameters used for the numerical
simulations. Vent height is the elevation of the base of the column above sea
level. The values

Now, as the

We applied the model to three different test cases with different vent and
atmospheric conditions:

test case 1 – low-flux plume without wind;

test case 2 – low-flux plume with wind (weak bent plume);

test case 3 – high-flux plume (strong plume).

The parameters used for the different test cases are listed in Table

For all the runs presented here, a single family of particles has been used,
with a normal distribution (with parameters

We first present a comparison of the plume profiles obtained with the three
different descriptions presented in the previous sections and highlighted in
the three colored boxes of Fig. 2 for the test case 2: method of moments for
the particle number that is the function of the size expressed in meters; method of
moments for the particle mass fraction that is the function of the size expressed
in the

Visualization of a normal initial distribution in the Krumbein

Atmospheric profiles for the three test cases. The height is expressed in meters above sea level, and for all the test cases the vent is located at 1500 m above sea level. For the wind profiles, only the profiles for the two test cases with wind are plotted.

In this section we want to study the variation during the ascent of solid
mass flux (due to particle settling) and of the mean and the variance of the
mass distribution along the column. As shown in the previous section, there
are no significant differences in the results obtained with the three
different descriptions of the grain-size distribution. For this reason, in
the following we restrict the analysis only to the formulation based on the
moments of the mass fraction distribution as a function of the diameter
expressed in the

Height vs. radius (left) and velocity (right) for a low-flux plume, simulated with three different models. In blue the profiles obtained using 13 bins, in red the profiles obtained using a continuous distribution of the particle number density and in green using a continuous distribution of the mass fraction.

In Fig.

When dealing with volcanic processes and volcanic hazards, our understanding of the physical system is limited, and vent parameters (volatile contents, temperature, grain-size distribution, etc.) are often not well constrained or are constrained with significant uncertainty. These factors mean that it is difficult to predict the characteristic of the ash cloud released from the volcanic column with certainty. An alternative is to quantify the probability of the outcomes (for example the grain-size distribution at the top of the column) by coupling deterministic numerical codes with stochastic approaches. It is our goal in this work also to assess the ability to systematically quantify the uncertainty and the sensitivity of the plume model outcomes to uncertain or variable input parameters, in particular to those characterizing the grain-size distribution at the base of the eruptive column.

Particle distribution parameters (mean, variance and skewness) and cumulative loss of solid mass flux for the test case 2 (low flux without wind), simulated with the formulation based on the moments of the mass fraction distribution.

Uncertainty quantification (UQ) or nondeterministic analysis is the process
of characterizing input uncertainties, propagating forward these
uncertainties through a computational model, and performing statistical or
interval assessments on the resulting responses. This process determines the
effect of uncertainties on model outputs or results. In particular, in this
work we wanted to investigate for different test cases the uncertainty in
four response functions (plume height, solid mass flux lost and mean
and variance of the mass fraction distribution at the top of the eruptive
column) when the mean and the standard deviation of the distribution at the
base are random variables with a uniform probability distribution in the
space

In volcanology Monte Carlo simulations are frequently used to perform
uncertainty quantification analysis. These methods rely on repeated random
sampling of input parameters to obtain numerical results; typically one runs
simulations many times over in order to obtain the distribution of an unknown
output variable. The cost of the Monte Carlo method can be extremely high in
terms of number of simulations to run, and thus several alternative approach
have been developed. LHS is another sampling technique
for which the range of each uncertain variable is divided into

Two-parameters Latin hypercube sampling (LHS) with 10 points (left) and
tensor product grid using

An alternative approach to uncertainty quantification is the so-called
generalized polynomial chaos expansion method, a technique that
mirrors deterministic finite element analysis utilizing the notions of
projection, orthogonality and weak convergence

We present here the results of several tests performed coupling the plume
model with the Dakota toolkit

As mentioned previously, the aim of the gPCE is to express the output of the
models as polynomials and these polynomials can be used to obtain response
surfaces for the output parameters as functions of the unknown input
parameters through the polynomials defined by Eq. (

Cumulative distributions and response surfaces for test case 1
(low-flux plume without wind). In the top panels the cumulative probability
for several variables describing the outcomes of the simulations (mean and
variance of the grain-size distribution at the top of the column, column
height and cumulative fraction of solid mass lost) are plotted for the
uncertainty quantification analysis carried out with the two different
techniques and for different numbers of simulations. The contour plots of the
response functions of the four output variables, resulting by the polynomials
given by Eq. (

In Fig.

Response surfaces for test case 2 (low-flux plume with wind, four top panels) and test case 3 (strong plume with wind, four bottom panels) obtained with the PCE with 81 quadrature points. Please note that the color scale is not consistent between plots.

With the polynomial chaos expansion it is also possible to easily obtain the
variance-based sensitivity indices

The results of the sensitivity analysis for the four outputs and the three
test cases investigated are presented in the bar plot of Fig.

Sobol main sensitivity indices. For each of the four output parameters the three bars are for the different test cases: test case 1 on the left, test case 2 in the middle and test case 3 on the right. For each test case the different colors of the bars are for the different sensitivity indices: blue for first-order sensitivity index with respect to the bottom TGSD mean, green for the first-order sensitivity index with respect to the bottom TGSD standard deviation and brown for the second-order combined sensitivity index.

In this work we have presented an extension, based on the method of moments,
of the Eulerian steady-state volcanic plume model presented in

An uncertainty quantification analysis has also been applied to the formulation based on the moments of the mass distribution. The results show, for the range of conditions investigated here and neglecting likely relevant interparticle processes such as particle aggregation and comminution, a small change of the mean and variance of the particle mass distribution along the column, indicating that the total grain-size distribution at the base of the vent represents a reasonable approximation of that at the top of the column. Furthermore, based on the plume model assumptions and outcomes, we observe a small sensitivity of the plume height to the initial grain-size distribution, with variations on the order of tens of meters for a plume rising to several kilometers.

For the application presented in this work, involving only two parameters, the comparison between the Latin hypercube sampling technique and the gPCE method shows that the latter only requires 81 simulations to produce the same results, in terms of cumulative probability distributions of several output, obtained with 1000 simulations and the LHS. In fact, the full uncertainty quantification analysis performed on a high-performance computing 48-multicore shared-memory system (HPC-SM) at Istituto Nazionale di Geofisica e Vulcanologia (INGV) in Pisa, Italy, required less than 2 s for the gPCE method with 81 quadrature points. These results make the new numerical code presented here, coupled with the uncertainty technique investigated, well-suited for real-time hazard assessment.

The source code with the input files for some simulation presented in this
work are available for download on the Volcano Modelling and Simulation
gateway (

The authors are grateful to Samantha Engwell for the helpful discussion and suggestions on the application of the model, and to L. Mastin, Y. Suzuki and M. Woodhouse for their thorough reviews of the manuscript. This work has been partially supported by the project MEDiterranean SUpersite Volcanoes (MED-SUV) FP7 ENV.2012.6.4-2 grant agreement no. 308665 (European Community). Edited by: J. Williams