Introduction
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
.
For instance, in most of the existing conduit models, crystals and bubbles
are treated as simple flow components and described by volume fractions only,
while in plume dynamics and ash dispersal models the grain-size distribution
(GSD) of pyroclasts is discretized in a finite number of classes (i.e., phases).
Both approaches make proper treatment of the continuous variability of the
dimension of pyroclastic particles and gas bubbles difficult. Literature
results clearly show that this
variability can largely affect relevant physical/chemical processes that
occur during the transport of the dispersed phase such as, for example, the
nucleation and growth of bubbles and the coalescence/breakage of bubbles and
crystals in the conduit or the aggregation of pyroclastic particles in a
volcanic plume.
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
, to track the evolution of these systems
not only in the physical space but also in the space of properties of the
dispersed phase (called internal coordinates). According to this method, a
population balance equation is formulated as a continuity statement written
in terms of a density function. From the density function some integral
quantities of interest (namely, the moments, i.e., specific quantitative
measures of the shape of the density function) are then derived and their
transport equations are formulated.
In this work we present an extension of the Eulerian steady-state volcanic
plume model presented in (derived from
) obtained by adopting the method of moments. In
contrast to the original works where pyroclastic particles are partitioned
into a finite number of classes with different sizes and properties, the new
model is able to consider a continuous size distribution function of
pyroclasts, f(D), representing the number or the mass fraction of particles
(per unit volume) with diameter between D and D+dD. Accordingly,
conservation equations of the plume are expressed in terms of the transport
equations for the moments of the ash particle size distribution. In
particular, in the following we present the new multi-phase model formulation
based on the implementation of the quadrature method of moments
and we investigate the sensitivity of the model to
uncertain or variable input parameters such as those describing the
grain-size distribution of the mixture. To quantify and incorporate this
epistemic uncertainty affecting the input parameters (characterizing
lack-of-knowledge) into our application of the model, we tested two different
approaches, a modification of the Monte Carlo method based on Latin hypercube sampling (LHS) and a stochastic approach, namely, the generalized polynomial chaos expansion (gPCE) method.
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.
Method of moments
Moments of the size distribution
In contrast to previous works, where the solid particles are partitioned in a
finite number of classes with different sizes , we
introduce here a continuous size distribution function representing the
number (or mass) concentration of particles (per unit volume) as a function
of particle diameter. In general, this particle size distribution (PSD)
is a function of time t, of the spatial coordinate and of the diameter of
the particles.
First, we present the method of moments for a particle size distribution
f(D), representing the number concentration of particles (particles per
unit volume) with diameter between D and D+dD, where D is expressed in
meters. When more than one family of particles are present, for example
lithics and pumice, we will use the subscript j to distinguish among them.
Consequently, the function fj(D) will denote the number concentration of
particles of the jth family.
Given a particle size distribution fj(D), we observe that its “shape”
can be quantified through the moments Mj(i) ,
defined by
Mj(i)=∫0+∞Difj(D)dD.
The particular definition of fj(D) we adopt, expressing the number
concentration of particles of size D, allows for the following physical
interpretation of the first four moments:
Mj(0) is the total number of particles of the jth family (per unit
volume).
Mj(1) is the sum of the particle's diameters of the jth family (per unit
volume).
Mj(2) is the total surface area of particles of the jth family (per unit
volume).
π6Mj(3) is the total volume of particles of the jth family (per unit volume)
or the local volume fraction of the jth dispersed phase, also denoted with αs,j. The
multiplying factor π6 is obtained assuming spherical particles. For particles with
different shape, if volume scales with the third power of length, we can still relate the
particle volume V with the particle length D through a volumetric shape factor kv such as V=kvL3.
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. ), and in this way it is possible to relate the
moments of the distribution with the mean, variance, skewness and kurtosis.
Furthermore, a mean particle size can be defined as the ratio of the moments
Mj(i+1)/Mj(i) for any value of i. For example, the Sauter mean
diameter (defined as the ratio between the mean volume and the mean surface
area) is obtained by setting i=2, giving Lj,32=Mj(3)/Mj(2).
Similarly, it is possible to define the mean particle length averaged with
respect to particle number density Lj,10=Mj(1)/Mj(0), i.e., the
sum of the lengths of particles (per unit volume) divided by the number of
particles (per unit volume), and the mean particle length averaged with
respect to particle volume fraction Lj,43=Mj(4)/Mj(3).
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
. The moments approach also allows one to treat
interparticle processes such as particle aggregation and fragmentation that
strongly depend on and affect the GSD of the mixture .
The moments and the corresponding transport velocities appear naturally in
the mathematical formulation as a direct consequence of the integration of
the Eulerian particle equations over the diameter spectrum, as will be shown
in the next section.
Moments of other quantities
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 fj(D). In general, for a quantity
ψj that is a function of the diameter D, we define its moments as
ψj(i)=1Mj(i)∫0+∞ψj(D)Difj(D)dD.
As a first example, we consider here the moments of particle density
ρs. In particular, following ,
density of lithics is assumed to be constant, whereas density of pumices
ρs,pum(D) with diameter D<D2 (here equal to 2 mm) is assumed to
decrease and to reach the lithic density value when the fragment diameter
decreases below D1 (here equal to 8 µm). Substituting the expression
for the particle density of the jth particle family in Eq. (), we obtain the moments of the density as
ρs,j(i)=1Mj(i)∫0+∞ρs,j(D)Difj(D)dD.
We remark that moments of different order are generally different, they will
only be equal (ρs,j(l)=ρs,j(m), l≠m) in two
limiting cases: for a monodisperse distribution with diameter D* and
density ρs*, i.e., fj(D)=δ(D-D*) (where δ is the
Dirac-delta function) and ρs,j(D*)=ρs*; or if all particles
have the same density, i.e., ρs,j(D)=ρs,j*, ∀D. In
both cases, ρs,j(i)=ρs,j*, ∀i. Otherwise, there is
no reason, e.g., for ρs,j(1) and ρs,j(3) to be the
same, as they represent length- and volume-weighted density averages,
respectively. For our application, we are interested mostly in the volumetric-averaged density ρs,j(3), i.e., the average mass per unit volume of
particles from now on denoted with ρ̃s,j.
The moments defined by Eq. () can also be used to define
other properties of the gas-particles mixture. For example, it follows from
the definition of the moments that if we have a mixture of a gas with density
ρg and a family of polydisperse distributions of particles with density
ρs,j=ρs,j(D), the mixture density is given by
ρmix=∑jαs,jρ̃s,j+(1-∑jαs,j)ρg=∑jπ6Mj(3)ρs,j(3)+(1-∑jπ6Mj(3))ρg,
and consequently the mass fraction of the jth solid phase with respect to
the gas-particles mixture is given by
xs,j=αs,jρ̃s,jρmix=π6Mj(3)ρs,j(3)∑jπ6Mj(3)ρs,j(3)+(1-∑jπ6Mj(3))ρg.
We also remark here that the gas phase is a mixture of atmospheric air,
entrained in the plume during the rise in the atmosphere, and a volcanic gas
component, generally water vapor. In the following, we will use the
subscript atm to denote the atmospheric air and wv for the volcanic water
vapor.
In contrast to the approach used in , where a
constant settling velocity for each class is provided by the user, here
several models have been implemented in the code
. For the application
presented in this work, the settling velocity is defined as a function of the
particle diameter and density as in :
ws,j(D)=k1D22ρs,j(D)ρatm0ρatmD≤10µmk2D2ρs,j(D)ρatm0ρatm10<D≤103µmk3D2ρs,j(D)CDρatm0ρatmD>103µm,
where k1=1.19×105 m2 kg-1 s-1, k2=8 m3 kg-1 s-1 and k3=4.833 m2 kg-1/2 s-1. The drag
coefficient CD is a parameter accounting for the particle surface
roughness, and for this work we used a value of 0.75 as in .
As carried out for particle density, it is possible to evaluate the
moments ws,j(i) of the settling velocity ws,j(D), defined as
ws,j(i)=1Mj(i)∫0+∞ws,j(D)Difj(D)dD
and representing weighted integrals of the settling velocity over the size
spectrum. Again, moments of different order are generally different. There is
no reason, e.g., for ws,j(2) and ws,j(3) to be the same, as
they represent surface and volume-weighted averages, respectively.
Finally, it is possible to define the moments Cs,j(i) of the
particles' specific heat capacity Cs,j:
Cs,j(i)=1Mj(i)∫0+∞Cs,j(D)Difj(D)dD.
We observe that for the specific heat capacity, we are generally not
interested in a volumetric average but in the mass average, denoted here with
the notation C‾s,j and given by the following expression:
C‾s,j=∫0+∞Cs,j(D)ρs,j(D)D3ρ̃s,jMj(3)fj(D)dD=1ρ̃s,jCs,jρs,j(3).
Mass fraction distribution
While in chemical engineering, where the method of moments is commonly used,
the particle number distribution fj(D) is generally preferred to describe
the polydispersity of the particles; in volcanology it is more common to use
a mass fraction distribution γj(ϕ), defined as a function of the
Krumbein phi (ϕ) scale:
ϕ=-log21000DD0,
where D is the diameter of the particle expressed in meters, and D0 is
a reference diameter, equal to 1 mm (to make the equation dimensionally
consistent).
In this case, the distribution γj(ϕ) represents the mass fraction
of particles (mass per unit mass of the gas-particles mixture) of the jth
family with diameter between ϕ and ϕ+dϕ. Again, the shape of the
distribution γj(ϕ) can be characterized by its moments
Πji, defined by
Πj(i)=∫-∞+∞ϕiγj(ϕ)dϕ.
Also in this case the particular definition of γj(ϕ) allows for a
physical interpretation of the moments; for example, the moment Πj(0)
is the mass fraction of the jth solid phase xs,j with respect to the
gas-particles mixture. As carried out for particle number distribution, it is
possible to define a mean particle size in terms of the moments of the mass
fraction distribution as Πj(i+1)/Πj(i); this ratio, for i=0,
gives the mass-averaged diameter, corresponding to the volume-averaged
diameter Lj,43=Mj(4)/Mj(3) when the density ρs,j(ϕ)
is constant.
Again, it is possible to define the moments of other quantities
ψj(ϕ) in terms of the continuous distribution of mass fraction
γj(ϕ) as
ψj(i)=1Πj(i)∫-∞+∞ψj(ϕ)ϕiγj(ϕ)dϕ.
For example, when the mass fraction distribution γj(ϕ) is used,
the mass-averaged heat capacity C‾s,j is given by the following
expression:
C‾s,j=1xs,j∫-∞+∞Cs,j(ϕ)γs,j(ϕ)dϕ=Cs,j(0)
and the volumetric-averaged density, i.e., the mass of particles per unit
volume, can be evaluated from
1ρ̃s,j=1xs,j∫-∞+∞γs,j(ϕ)ρs,j(ϕ)dϕ=1ρs,j(0).
Plume model
In this section we describe the assumption and the equations of the model. As
in , the model assumes an homogeneous mixture of
particles and gases with thermal and mechanical equilibrium between all
phases. Aggregation and breakage effects are not considered and consequently
density does not change with time. Finally, the model does not consider
effects of humidity and water phase changes.
The equation set for the plume rise model is solved in a 3-D coordinate
system (s,η,θ) by considering the bulk properties of the eruptive
mixture (see Fig. ). The plume is assumed with a circular
section in the plane normal to the centerline trajectory with curvilinear
coordinate s, a top-hat profile of the velocity along the centerline, an
inclination on the ground defined by an angle η between the axial
direction and the horizon, and an angle θ in the horizontal plane
(x,y) with respect to the x axis. These angles are needed to describe the
evolution of weak explosive eruptions that are strongly affected by
atmospheric conditions.
Schematic representation of the Eulerian plume model. The dashed
black line represent the axis of the curvilinear coordinate s.
Following and , the conservation of flux
of particles with size D of the jth family is given by
ddsfj(D)πr2Usc=-2πrpws,j(D),fj(D)
where r is characteristic plume radius, Usc represents the velocity of
the plume cross section along its centerline (a top-hat profile is assumed)
and p is the probability that an individual particle will fall out of the
plume, defined as a function of an entrainment coefficient α as
p=1+65α2-11+65α2+1.
Equation () states that the number of particles of the
jth family with size D lost from the plume is proportional to the number
of particles at the plume margin, given by fj(D)×2πr, to the
settling velocity ws,j(D) and to the probability factor p.
Now, multiplying both the sides of Eq. () for
Di and then integrating over the size spectrum [0,+∞], we obtain
the following conservation equations for the moments Mj(i):
ddsMj(i)Uscr2=-2rpws,j(i)Mj(i).
If we compare our formulation with that presented in ,
where the effects of a polydisperse solid phase are taken into account
partitioning the size spectrum in a finite number N of solid classes, the
set of Eq. () replaces the N mass conservation
equations for the N particulate classes.
From Eq. (), if we multiply both the terms by the
mass of the particles of size D, given by π6D3ρs,j(D),
we obtain the additional equation
ddsfj(D)π6D3ρs,j(D)πr2Usc=-2πrpws,j(D)fj(D)π6D3ρs,j(D)
and, integrating over the size spectrum,
ddsUscr2π6Mj(3)ρs,j(3)=-2rpπ6M(3)ws,jρs,j(3),
where on the left-hand side the term π6Mj(3)ρs,j(3) represents the volume average bulk density of the particles
of the jth family (i.e., the mass of particles of the jth family per
unit volume of gas-particles mixture, denoted with the superscript B,
ρs,jB), while on the right-hand side the term ws,jρs,j(3) represents the mass-averaged settling velocity of
the particles of the jth family multiplied by the volume-averaged
particle density. Equation () is the mass conservation
equation for the jth family of particles, relating the variation of the
mass flux of particles within the plume with the loss at the plume margin.
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, αϵ and
γϵ. The theory assumes that the efficiency of mixing with
ambient air is proportional to the product of a reference velocity (the
vertical plume velocity in one case and the wind field component along the
plume centerline in the other), by αϵ and γϵ
. Thus,
following and , we define the
entrainment velocity Uϵ as a function of wind speed, Uatm, as
well as axial plume speed, Usc:
Uϵ=αϵ|Usc-Uatmcosϕ|+γϵ|Uatmsinϕ|,
where αϵ|Usc-Uatmcosϕ| is entrainment by radial
inflow minus the amount swept tangentially along the plume margin by the
wind, and γϵ|Uatmsinϕ| is entrainment from wind.
With this notation, the total mass conservation equation solved by the model
becomes
ddsρmixUscr2=2rρatmUϵ-2rp∑jπ6Mj(3)ws,jρs,j(3),
stating that the variation of mass flux (left-hand-side term) is due to air
entrainment (first right-hand-side term) and loss of solid particles (second
right-hand-side term), as obtained from Eq. ().
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
ddsρmixUscr2(u-Uatm)=-r2ρmixwdUatmdz-2upr∑jπ6Mj(3)ws,jρs,j(3),
and
ddsρmixUscr2w=gr2(ρatm-ρmix)-2wpr∑jπ6Mj(3)ws,jρs,j(3),
where the two components of plume velocity along the horizontal and vertical
axes are u and w, respectively, and they are linked by the relation
Usc=u2+w2. In the right-hand side of Eq. () the terms related to the exchange of momentum due
to the wind and to momentum loss from the fall of
solid particles appear. Similar contributions are evident in the right-hand-side term of Eq. () where the vertical momentum
is changed by the gravitational acceleration term and the fall-out of
particles.
Now, following the notation adopted above and denoting with T the mixture
temperature, the equation for conservation of thermal energy solved by the
model is written as
ddsρmixUscr2CmixT=2rρatmUϵCatmTatm-r2wρatmg-2Tpr∑jπ6Mj(3)Cs,jws,jρs,j(3).
The first term on right-hand side describes the cooling of the plume due to
ambient air entrainment, the second one takes into account atmospheric
thermal stratification, and the third term allows for heat loss due to loss
of solid particles. Again, this last term is obtained writing the heat loss
for the particles of size D, and then integrating over the size spectrum. A
thermal equilibrium between solid and gaseous phases is assumed. In Eq. () Catm and Cmix are the heat capacity of the
entrained atmospheric air and of the mixture, respectively, the latter being
defined as
Cmix=1-∑jxs,jCp,g+∑jxs,jC‾s,j
or, in terms of the bulk densities ρatmB=xatmρmix,
ρwvB=xwvρmix and ρs,jB=π6Mj(3)ρ̃s,j, as
Cmix=ρatmBCatm+ρwvBCwv+∑jρs,jBC‾s,jρatmB+ρwvB+∑jρs,jB.
From this expression, if we multiply all the terms at the numerator and the
denominator of the right-hand side by Uscr2 and we differentiate with
respect to s, we obtain after some cancellation and algebra manipulations
the following equation for the variation of the mixture specific heat with
s:
dCmixds=1ρmixUscr2Catm-CmixddsρatmBUscr2+∑jC‾s,j-Cmixddsρs,jBUscr2+∑jρs,jBρmixddsC‾s,jρs,jBUscr2ρs,jBUscr2-C‾s,jddsρs,jBUscr2ρs,jBUscr2.
Now, substituting the expressions for the derivatives appearing in each term
on the right-hand side, we obtain the equation for the variation rate of
mixture specific heat in terms of the moments:
dCmixds=1ρmixUscr2[Catm2rρatmUϵ-Cmix(2rρatmUϵ-2pr∑jπ6Mj(3)ws,jρs,j(3))-2pr∑jπ6Mj(3)ws,jρs,jCs,j(3)].
Similarly, a gas constant Rg can be defined as a weighted average of the
gas constant for the entrained atmospheric air Ratm and the gas constant
of the volcanic water vapor Rwv
Rg=ρatmBRatm+ρwvBRwvρatmB+ρwvB,
and a conservation equation can be derived, knowing that the variation of
gaseous mass fraction with height is solely due to entrained air:
dRgds=Ratm-Rgρmix(1-xs)Uscr2×2rρatmUϵ.
This formulation reduces, for particular cases, to the expressions of
and . Equations ()
and () are needed in order to close the system of equations and
recover the new values of the temperature and the mixture density once the
system of ordinary differential equations is integrated. Otherwise, without
the solutions of Eqs. () and (), we should use
the old values of ρmix and Cmix at s to obtain the values of
the temperature at s+ds from the lumped term
(ρmixUscr2CmixT) obtained integrating Eq. ().
Finally, as in , the equations expressing the
coordinate transformation between (x,y,z) and (s,η,θ) are given
by
dzds=sinη,dxds=cosηcosθ,dyds=cosηsinθ.
Mass fraction distribution
Similarly to the distribution of particle number fj(D) and the
moments Mj(i), it is possible to derive a set of conservation equations
in terms of the moments Πj(i) of the mass fraction distribution
γj(ϕ) expressed as a function of the Krumbein scale.
In this case, the conservation of mass flux of particles with size ϕ of
the jth family is written as
ddsρmixγj(ϕ)πr2Usc=-2πrpws,j(ϕ)ρmixγj(ϕ).
Multiplying both sides of the equation by ϕi and integrating over the
size spectrum [-∞,+∞], we obtain the following conservation
equations for the moments of the continuous distributions γj(ϕ):
ddsΠj(i)ρmixUscr2=-2rpρmixws,j(i)Πj(i).
For i=0, the equations of conservation of the moments give
ddsxs,jρmixUscr2=-2rpρmixws,j(0)xs,j
expressing the loss of mass flux of the particles of the jth family and
thus we can write the total mass conservation equation as
ddsρmixUscr2=2rρatmUϵ-2rpρmix∑jws,j(0)Πj(0).
From the variation of mass flux, as was carried out for the distribution of particle
number fj(D) and the moments Mj(i), we derive the horizontal and
vertical components of the momentum balance:
ddsρmixUscr2(u-Uatm)=-r2ρmixwdUatmdz-2uprρmix∑jws,j(0)Πj(0),
ddsρmixUscr2w=gr2(ρatm-ρmix)-2wprρmix∑jws,j(0)Πj(0).
The equation for conservation of thermal energy is
ddsρmixUscr2CmixT=2rρatmUϵCatmTatm-r2wρatmg-2Tprρmix∑jCs,jws,j(0)Πj(0)
and the equation for the variation rate of mixture specific heat in terms of
the moments of the mass fraction distribution is written as
dCmixds=1ρmixUscr2[Catm2rρatmUϵ-Cmix(2rρatmUϵ-2rpρmix∑jws,j(0)Πj(0))-2prρmix∑jCs,jws,j(0)Πj(0)].
The formulation of the equations for the gas constant Rg and the
coordinates of the (x,y,z) remain unchanged.
Numerical scheme
The plume rise equations are solved with a predictor–corrector Heun's scheme
that guarantees a second-order accuracy, keeping
the execution time on the order of seconds. If we rewrite the system of
ordinary differential equations with the following compact notation:
dyds=f(s,y),y(s0)=y0,
where y is the vector of the quantities in the left-hand sides of the
conservation equations presented in the previous section, then the procedure
for calculating the numerical solution by way of Heun's method
is to first calculate the intermediate values
ỹi+1 and then the solution yi+1 at the next integration
point
ỹi+1=yi+dsf(si,yi),predictor stepyi+1=yi+ds2f(si,yi)+f(si+1,ỹi+1),corrector step.
Quadrature method of moments
We observe that to calculate the right-hand side for both the predictor and
corrector step we need not only the moments M(i) but also the
additional moments [ws]i, [wsρs](i) and [wsρsCs](i). As in , the integral in the
definition of these moments is replaced by a quadrature formula and the
moments, for a generic variable ψ=ψ(D), are approximated as
ψ(i)=1M(i)∫0+∞ψ(D)f(D)DidD≈∑l=1Nψ(Dl)Dliωl.
Here ωl and Dl are known as “weights” and “nodes” (or
“abscissae”) of the quadrature, respectively, and the accuracy of a
quadrature formula is quantified by its degree. The degree of accuracy is
equal to d if the interpolation formula is exact when the integrand is a
polynomial of an order less than or equal to d and there exists at least one
polynomial of an order d+1 that makes the interpolation formula inexact. In
particular, an N point Gaussian quadrature rule, is a quadrature rule
constructed to yield an exact result for polynomials of degree 2N-1 or less
by a suitable choice of the nodes Dl and weights ωl for
l=1,…,N .
The Wheeler algorithm, as presented in ,
provides an efficient O(N2) algorithm for finding a full set of weights
and abscissas for a realizable moment set. The resulting nodes Dl are
always within the support (and therefore represent realizable values of the
particle size), and the weights ωl are always positive, ensuring
that, when the quadrature is used, accurate results are obtained
. Nevertheless, these properties are
respected only if the moment set is realizable, meaning that there exists a
particle size distribution resulting in that specific set of moments.
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 . The algorithm first verifies whether the moment
set is realizable (by looking at the second-order difference vector or by
looking at the Hankel–Hadamard determinants; ). If the
moment set is unrealizable it proceeds with the correction. In the numerical
model presented here, the implementation of the correction algorithm of
Wright (1984) is derived from the version presented in
.
Thus, in both the predictor and corrector step, the following algorithm is
used:
The nodes Dj,l and weights ωj,l are calculated with the
Wheeler algorithm for l=1,…,N.
The quadrature formula (Eq. ) is used to evaluate the
moments [ws]j(i), [wsρs]j(i) and [wsρsCs]j(i).
The right-hand side of the ODE's system (Eq. ) is evaluated
explicitly.
The solution is advanced with the predictor (or the corrector) step of
the Heun's scheme.
For each particle family j, the moments Mj(i) (i=0,…,2N-1),
if required, are corrected with the McGraw (or Wright) algorithm.
We observe that if the jth family of particles is monodisperse with
diameter d‾j, the Wheeler algorithm fed with the first two moments
only gives a result of a single quadrature node Dj,1=d‾j with weight
ωj,1=1. This allows us also to use the model for the simplified
case where the solid particle distribution is partitioned in a finite number
of classes with constant size, assigning to each class a monodisperse
distribution.
Initial condition
Initial conditions at the vent include the initial plume radius (r0),
mixture velocity (Usc,0) and temperature (T0), gas mass fraction
(n0) and the particle size distribution through the initial moments
M0(i). In the next section we derive analytically the moments of a
specific initial distribution (a normal distribution in the Krumbein scale)
for both the formulations based on the number of particles as a function of
the particle diameter expressed in meters and the formulation based on the
mass concentration expressed as a function of the phi scale.
Lognormal distribution
For the application presented in this work, the initial distribution f(D)
at the base of the plume is defined as a function of the particle diameter
expressed in meters (m), in order to give a corresponding normal
distribution with parameters μ and σ for the mass concentration
expressed as a function of the Krumbein phi (ϕ) scale (when all the
particles have the same density):
γ(ϕ)=K0σ2πe-(ϕ-μ)22σ2,
where K0 is a parameter that has to be chosen in order to satisfy the
initial condition on the solid mass fraction.
Given the parameters μ and σ, the initial distribution f(D) is
then written as
f(D)=6C0-σln2D42π3e--ln(1000D)-μln222σln22,
where C0, analogously to K0, is a parameter that has to be fixed in
order to satisfy the initial condition prescribed for the mass (or volume)
fraction of particles.
We observe that if we introduce the following re-scaled variables for the
diameter, the mean and the variance:
D‾=1000D,μ‾=-μln2,σ‾=-σln2,
then it is possible to rewrite the particle distribution f(D) in terms of a
lognormal distribution in the variable D‾ with parameters μ‾
and σ‾:
f‾(D‾)=6×1012C0πD‾31σ‾D‾2πe-ln(D‾)-μ‾22σ‾2=6×1012C0πD‾3lognorm(D‾,μ‾,σ‾).
Consequently, we can evaluate the moments M(i) of f(D) analytically
from the moments of the lognormal distribution as
M(i)=6C0π103(3-i)exp(i-3)μ‾+12(i-3)2σ‾2,
and we obtain, for the third moment,
M(3)=6C0π⇒C0=αs0,
where αs0 is the initial volume fraction of the particles in the
solid-gas mixture.
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 DA expressed in meters can be
evaluated as
DA=L32=M(3)M(2)=10-3expμ‾-12σ‾2,
or, if expressed in ϕ, as
DAϕ=L32ϕ=μ+12σ2ln(2).
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 μ of the lognormal distribution by a factor proportional to the
variance σ2. For numerical models describing the multi-phase
(particulate) nature of the matter and which approximate the particle size
distribution with an average size, it is hence more appropriate to use, as
particle size representative of a lognormal distribution, the Sauter mean
diameter than the mean diameter μ. The difference between the two
approximations is smaller the narrower the particle size distribution. We
must also remark that, for particles in the inertial-dominated regime
(e.g., Rep>2000), showed that the Sauter mean diameter is
the effective diameter, regardless of particle shape, particle size
distribution, particle density distribution or net volume fraction; for
particles in the creeping flow regime (Rep≪1) the effective mean diameter
is the volume-width diameter.
When the Sauter mean diameter is used, also the variance and the standard
deviation SD should be based on the specific surface area
. Hence,
σA2=∫0+∞1D-1DA2π6D3f(D)dD,
or expressed as a function of the moments:
σA2=M(1)M(3)-(M(2))2(M(3))2.
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 L32=M(3)/M(2) and
the mean particle length averaged with respect to particle number density
L10=M(1)/M(0) (or if the first 4 moments) are known, then we can
solve for the re-scaled mean and variance μ‾ and σ‾ the
following system:
L10=10-3expμ‾-52σ‾2L32=10-3expμ‾-12σ‾2.
Once the re-scaled mean and variance are known, we can obtain μ and
σ in the Krumbein ϕ scale.
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 μ and variance
σ2, then the continuous distribution is given by Eq. (). We observe that this expression of the distribution
is not based on the assumption of constant density for the particles of
different size.
In this case, the moments Π(i) are given by the following expression
Π(i)=K0∑j=0⌈i/2⌉i2j(2j-1)!!σ2jμi-2j.
where the symbols ⌈⌉ and !! denote the integer part and the
double factorial (n!!=∏k=0m(n-2k), where m=⌈n/2⌉-1),
respectively.
Input parameters used for the numerical
simulations. Vent height is the elevation of the base of the column above sea
level. The values ρ1,2 and D1,2 are used to compute the density
of the particles as a function of the diameter, according to the formulation
of . The values reported for μ and
σ define the range used for the uncertainty quantification and
sensitivity analysis.
Parameters
Units
Test case 1
Test case 2
Test case 3
Vent radius
m
27
27
708
Vent velocity
m s-1
135
135
275
Vent temperature
K
1273
1273
1053
Vent gas mass fraction
0.03
0.03
0.05
Vent height
m
1500
1500
1500
ρ1
kg m-3
2000
2000
2000
ρ2
kg m-3
2600
2600
2600
D1
m
8×10-6
8×10-6
8×10-6
D2
m
2×10-3
2×10-3
2×10-3
μ
ϕ
[-1.0;3.0]
[-1.0;3.0]
[-1.0;3.0]
σ
ϕ
[0.5;2.5]
[0.5;2.5]
[0.5;2.5]
Now, as the 0th moment is equal to the mass fraction of particles, we
obtain K0=xs. Furthermore, we observe that the mass fraction-averaged
diameter in the ϕ scale is given by the ratio Π(1)/Π(0),
while the variance of the mass fraction distribution can be evaluated as
Π(2)Π(0)-(Π(1))2/(Π(0))2. These two
quantities correspond to the parameters (μ,σ2) generally used to
describe the mass fraction when a normal distribution in the ϕ scale is
assumed. For this reason, when we want to track the changes of the mass
fraction-averaged diameter and its standard deviation (or variance) in the
ϕ scale during the plume rise, it is preferred to use a formulation
based on the moments Π(i) than the moments M(i).
Application
Simulation inputs
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 , while the atmospheric
conditions are plotted in Fig. . For the low-flux plumes, a mass flow rate of 1.5×106 kg s-1 has been fixed, while for the strong plume the value is 1.5×109 kg s-1. The temperature pressure and density profiles used for the test
case without wind (test case 1) are those defined by the International
Organization for Standardization for the International Standard Atmosphere
, while the profiles for the other two test cases come
from reanalysis data.
For all the runs presented here, a single family of particles has been used,
with a normal distribution (with parameters μ and σ) for the mass
concentration as a function of the diameter expressed in the ϕ scale and
with density varying with the particle diameter.
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 ϕ scale; discretization in uniform bins in the ϕ scale. For
this comparison, the mass flow rate at the vent is 1.5×106 kg s-1 and a
rotating wind is present, as shown in Fig. , while the mean
and the standard deviation of the initial total grain-size distribution are,
respectively, 2 and 1.5, expressed in the ϕ scale. The results of the
numerical simulations obtained with the three different formulations are
presented in Fig. and they perfectly match, showing that
the method of moments (dotted lines), both applied to the continuous
distribution of the particle number (red) or to the mass distribution
(green), gives the same results of the classical formulation based on the
discretization of the mass distribution in bins (solid line). For these
simulations, we used only the first six moments of the distributions, while 13 bins have been employed with the discretized formulation. This results in a
smaller number of equations to solve for the method of moments and, despite
the additional cost of the method of moments due to the evaluation of the
quadrature points and formulas through the Wheeler algorithm, in a smaller
computational time, with a gain of about 30 %.
Visualization of a normal initial distribution in the Krumbein
ϕ scale for the solid particles. On the top plot, the particle number
distribution expressed as a function of the diameter expressed in meters is
plotted. On the second and third plots from the top, the corresponding
distributions of volume and mass are plotted, these two being different
because the density is a function of the diameter. On the fourth plot the
continuous distribution (lognormal) of mass fraction as a function of the
ϕ scale is plotted, while in the last plot the distribution has been
discretized with 13 bins in the range (-4;8). On each panel different average
radii are also plotted, together with the mean of the initial distribution.
The first, fourth and fifth panel are highlighted with different colors, also
used in Fig. for the solutions obtained with the three
different representations of the initial grain-size distribution.
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.
Simulation results
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 ϕ scale. With this approach, the mean, the variance and
the skewness of the mass distribution along the column are easily obtained
from the first four moments Π(i) of the mass fraction distribution.
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. we present the results relative to the test case 2 for an initial particle size distribution with mean diameter 2 and standard
deviation 1.5, expressed in the ϕ scale. In the left and middle panels
the mean, the variance and the skew of the mass fraction distribution are
shown, while in the right panel the cumulative loss of solid
mass flux is plotted as a percentage of the initial value. We observe a
decrease in the mean size of the particles, due to the different settling
velocities of particles of different sizes. A decrease in the variance of the
size distribution with height is also observed from the second plot. We
remark that the particles have a normal distribution only at the base of the
column (resulting in a null skewness), and the negative skew at the top of
the column indicates that the tail on the left side of the grain-size
distribution is longer than the tail on the right side; i.e., the mass is more
concentrated on the right of the spectrum of particle sizes (finer
particles). For this reason we do not have to look at the mean and the
variance plotted in Fig. as the parameters of a normal
(and symmetric) distribution. Nonetheless, changes in the mean, the variance
and the skewness are observed, we remark that these changes are quite small
and for this reason the parameters of the total grain-size distribution at
the top of the eruptive column are a good approximation of the parameters at
the base of the column, and vice versa. However, this is true for the
specific input condition of this test case and not in general. For this
reason, it is important to quantify the uncertainty of this assumption for
different initial total grain-size distributions and different atmospheric
conditions.
Uncertainty and sensitivity analysis
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 (μ,σ)∈[-1;3]×[0.5;2.5].
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 Ns segments
of equal probability, where Ns is the number of samples requested. The
relative lengths of the segments are determined by the nature of the
specified probability distribution (e.g., uniform has segments of equal
width; normal has small segments near the mean and larger segments in the
tails). For each of the uncertain variables, a sample is selected randomly
from each of these equal probability segments. These Ns values for each of
the individual parameters are then combined in a shuffling operation to
create a set of Ns parameter vectors with a specified correlation
structure. Compared to Monte Carlo sampling, the LHS has
the advantage that in the resulting sample set every row and column in the
hypercube of partitions has exactly one sample, and thus a smaller number of
samples is required to cover all the parameter space. In the left panel of
Fig. an example of LHS with Ns=10
and a uniform distribution probability for both μ and σ is
plotted.
Two-parameters Latin hypercube sampling (LHS) with 10 points (left) and
tensor product grid using 9×9 Clenshaw–Curtis points (right).
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 . The polynomial chaos expansion (PCE) method was
developed by Norbert Wiener in 1938 and it soon became widely used because of
its efficiency when compared to Monte Carlo simulations. The term “chaos”
here simply refers to the uncertainties in input, while the word
“polynomial” is used because the propagation of uncertainties is described by polynomials.
If ζ is the vector of uncertain input variables, the aim of the gPCE is
to express the response function Y in the form of a polynomial ξ as
follows:
ξ(ζ)=ξ0+ξ1P1(ζ)+ξ2P2(ζ)+…+ξmPm(ζ),
where P1,…,Pm are polynomials that form an orthogonal basis. The
choice of the polynomials basis depends on the probability distribution of
the input variables. In particular, for a uniform distribution, the basis of
the expansion is given by the Lagrange polynomials. For the application
presented in this work the coefficients of the expansion have been evaluated
using a spectral projection where the computation of the required
multi-dimensional integrals is based on the tensor product of 1-D
Gaussian quadrature rules. In order to compute the quadrature points, the
grid used in our work is the Clenshaw–Curtis grid (Fig. ,
right), representing a good solution for a multi-dimensional Gaussian
quadrature with a small number of variables .
We present here the results of several tests performed coupling the plume
model with the Dakota toolkit to investigate systematically
the capability of the LHS and the gPCE techniques to assess 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 variance at the base are unknown. For all the test
cases three sets of 500, 1000 and 2000 simulations have been performed for
the LHS, and the results have been compared with those obtained with three
tests for the gPCE and respectively 9, 36 and 81 simulations performed for
the multi-dimensional quadrature. The first set of runs for the LHS,
consisting of 500 simulations only, was not sufficient to provide accurate
results and for this reason in the following we presents only the results
obtained with 1000 and 200 simulations. In order to compare the two
techniques, the cumulative distributions of the four response functions
obtained with the LHS and the gPCE, have been plotted in Fig. for test case 1 (no wind). On the x axes we can see
the range of the values obtained for the response functions: -1–3.5 for the
mean of the total grain-size distribution (TGSD) at top of the column expressed in the ϕ scale; 0.4–2.2
for the standard deviation; 10.41–10.47 km for the column height and
10–60 % for the percentage of solid mass flux lost. All the uncertainty
quantification tests produced very similar results, with a small difference
in the cumulative distribution observed only in the distribution of the solid
mass flux lost obtained with the gPCE technique and 9 and 36 quadrature
points. Similar results have been obtained for the other test cases (not
shown here). Thus, the results highlights that for the model and the
applications presented in this work gPCE represents a valid alternative to
Monte Carlo simulations, with a number of runs required to produce the same
accuracy reduced by a factor 10 (81 simulations vs. 1000 simulations). If more
parameters were varied, the computational cost would increase for both gPCE
and LHS, although the advantage of gPCE would be reduced.
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. (). In the
four bottom panels of Fig. the contours of the four
response surfaces for the output investigated in this work have been plotted,
showing the dependence on the uncertain input parameters. The mean and the
standard deviation of the TGSD at the top of the eruptive column clearly
reflects the corresponding values at the bottom, with a small effect of the
bottom standard deviation on the mean size at the top, resulting in an
increase in the average grain size with increasing values of the initial
standard deviation (see the curves in the first panel bending on the left for
higher values of σ). Conversely, the plume height for this test case
shows a nonlinear dependency but at the same time a small sensitivity to the
initial grain-size distribution, with changes, for the specific conditions
considered here, smaller than 1 % of the average height. This can be
explained by the fact that a large amount of air is entrained in the column
during the ascent and the contribution of the solid fraction to the overall
dynamics becomes small compared to that exerted by the gas. Finally, we
observe that the loss of particles is mostly controlled by the mean size of the
TGSD.
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. () and obtained with the PCE with 81 quadrature
points, are plotted in the bottom panels. The variables contoured in the
lower panels are the same as those on the horizontal axes in the upper
panels.
In Fig. the same contour plots are shown for the
polynomial expansion computed for test case 2 (top) and test case 3 (bottom)
with 81 quadrature points. The results show again that the total grain-size
distribution at the base of the vent represents a reasonable approximation of
that at the top of the column. For these test cases, both the column height
and the solid mass lost appear to be mostly controlled by the mean size of
the TGSD at the base of the column, with a small sensitivity of the height to
the initial grain-size distribution. We also observe that the maximum
percentage of loss in the solid mass flux is about 15 % for the strong plume
simulations, and it is attained for larger mean sizes and smaller variance of
the initial TGSD. This value is noticeably smaller than that obtained for the
weak bent test case (≈40 %) and for the weak test case without wind
(≈60 %). Despite the loss of particles, in both the cases the range
of variation of the column height is quite small and, as previously
mentioned, this is due to the large amount of air entrained in the volcanic
column that reduces the contribution of the solid fraction to the overall
dynamics. As an example to understand the relevance of the entrained air, for
a simulation performed for the low-flux plume without wind and with μ=2
and σ=1.5 in the ϕ scale, the mass flow rate at the top of the
column is 1.2×108 kg s-1, compared to the value at the base of
1.5×106 kg s-1.
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.
Sensitivity analysis
With the polynomial chaos expansion it is also possible to easily obtain the
variance-based sensitivity indices with no additional
computational cost. In contrast with some instances, where the term
sensitivity is used in a local sense to denote the computation of response
derivatives at a point, here the term is used in a global sense to denote the
investigation of variability in the response functions. In this context,
variance-based decomposition is a global sensitivity method that summarizes
how the variability in model output can be apportioned to the variability in
individual input variables . This sensitivity analysis uses two
primary measures, the main effect sensitivity index Si and the total
effect index Ti. These indices are also called the Sobol indices. The main
effect sensitivity index corresponds to the fraction of the uncertainty in
the output, Y, that can be attributed to input xi alone. The total
effects index corresponds to the fraction of the uncertainty in the output,
Y, that can be attributed to input xi and its interactions with other
variables. The main effect sensitivity index compares the variance of the
conditional expectation Varxi[E(Y|xi)] against the total variance
Var(Y). Formulas for the indices are
Si=Varxi(Y|xi)Var(Y)
and
Ti=E(Var(Y|X-i))Var(Y),
where Y=f(x) and x-i=(x1,…,xi-1,xi+1,…,xm). Similarly,
it is also possible to define the sensitivity indices for higher-order
interactions such as the two-way interaction Si,j. The calculation of
Si and Ti requires the evaluation of m-dimensional integrals that are
typically approximated by Monte Carlo sampling. However, in stochastic
expansion methods, it is possible to approximate the sensitivity indices as
analytic functions of the coefficients in the stochastic expansion.
The results of the sensitivity analysis for the four outputs and the three
test cases investigated are presented in the bar plot of Fig. . For each of the four groups (one for each of the different
output functions), the three bars represent the main sensitivity indices for
the three test cases (test 1 on the left, test 2 in the middle and test 3 on
the right), while the different colors are for the sensitivity indices with
respect to different variables (blue is for the mean of the initial TGSD,
green for the standard variation of the initial TGSD and brown for the second-order coupled interaction). Again, the sensitivity analysis confirms that the
mean and the standard deviation of the grain-size distributions at the top of
the eruptive column are controlled primarily by the respective parameters of
at base of the column. The mean of the TGSD also controls the percentage of
solid mass flux lost during the rise of the column and the plume height for
the two test cases with wind, while for the weak test case without wind the
dispersion of the distribution and second-order interaction also play a major
role in controlling plume height variability. However, as already observed
with the uncertainty quantification analysis, we remark that the variability
in the plume height, when the mean and the standard deviation of the TGSD
vary in the investigated ranges, is extremely small for all the test cases
(less than 1% with respect to the average values), and thus the investigation
of how the variability in model output can be apportioned to the variability
in individual input variables is less relevant for the plume height than for
the other output parameters.
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.
Conclusions
In this work we have presented an extension, based on the method of moments,
of the Eulerian steady-state volcanic plume model presented in
(derived from
). Two different
formulations, one based on a continuous distribution of the number of
particles as a function of the size and a second based on the continuous
distribution of the mass fraction, have been presented. The tracking of the
moments of mass distribution, defined as a function of the Krumbein phi
scale, has the advantage that with the first three moments only we are able
to recover the mean and the standard deviation of the total grain-size
distribution. The results of a comparison between the two formulations based
on the method of moments and the classical formulation based on the
discretization of the mass distribution in bins show that the different
approaches produce the same results, with an advantage of the method of
moments in terms of computational costs. Furthermore, a formulation based on
continuous description of particle size, is better suited to properly
describe complex interparticle processes such as particle aggregation and
fragmentation that are likely to play an important role in the plume
evolution. In particular, the method of moments has already been successfully
applied to model aggregation and breakage processes in particulate systems
.
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.
Code availability
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 (http://vmsg.pi.ingv.it/) and on the site for collaborative volcano
research and risk mitigation Vhub (https://vhub.org/).