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A simple semianalytical model to simulate ash dispersion and deposition produced by sustained Plinian and sub-Plinian eruption columns based on the 2D advection–dispersion equation was applied. The eruption column acts as a vertical line source with

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A model for the numerical simulation of tephra fall deposits
T. Pfeiffer
a,
*, A. Costa
b,1
, G. Macedonio
c
a
Department of Earth Sciences - University of Aarhus, Denmark
b
Dip. di Scienze della Terra e Geologico-Ambientali, Univ. di Bologna, Italy
c
Osservatorio Vesuviano INGV, Italy
Received 1 September 2003; accepted 2 September 2004
Abstract
A simple semianalytical model to simulate ash dispersion and deposition produced by sustained Plinian and sub-Plinianeruption columns based on the 2D advection–dispersion equation was applied. The eruption column acts as a vertical linesource with a given mass distribution and neglects the complex dynamics within the eruption column. Thus, the use of the model is limited to areas far from the vent where the dynamics of the eruption column play a minor role. Verticalwind and diffusion components are considered negligible with respect to the horizontal ones. The dispersion anddeposition of particles in the model is only governed by gravitational settling, horizontal eddy diffusion, and windadvection. The model accounts for different types and size classes of a user-defined number of particle classes andchanging settling velocity with altitude. In as much as wind profiles are considered constant on the entire domain, themodel validity is limited to medium-range distances (about 30–200 km away from the source).The model was used to reconstruct the tephra fall deposit from the documented Plinian eruption of Mt. Vesuvius, Italy,in 79 A.D. In this case, the model was able to broadly reproduce the characteristic medium-range tephra deposit. Theresults support the validity of the model, which has the advantage of being simple and fast to compute. It has the potential to serve as a simple tool for predicting the distribution of ash fall of hypothetical or real eruptions of a givenmagnitude and a given wind profile. Using a statistical set of frequent wind profiles, it also was used to construct air fallhazard maps of the most likely affected areas around active volcanoes where a large eruption is expected to occur.
D
2004 Elsevier B.V. All rights reserved.
Keywords:
ash fall; settling velocity; computer model; Vesuvius 79 A.D.
1. Introduction
Armienti et al. (1988)andMacedonio et al.(1988)presented a numerical 3D model of volcanictephra fallout. The model assumed that sufficientlyfar from the vent, the eruption column can beregarded as a vertical line source and that the
0377-0273/$ - see front matter
D
2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jvolgeores.2004.09.001* Corresponding author.
E-mail addresses:
tpfeiffer@decadevolcano.net (T. Pfeiffer)
8
costa@ov.ingv.it (A. Costa)
8
macedon@ov.ingv.it (G. Macedonio).
1
Now at Osservatorio Vesuviano INGV, Italy.Journal of Volcanology and Geothermal Research 140 (2005) 273–294www.elsevier.com/locate/jvolgeores
motion of particles can be sufficiently described byatmospheric advection, diffusion, and settling bygravity. The model numerically solves the 3Dadvection–diffusion–sedimentation equation, assum-ing constant atmospheric eddy diffusion coeffi-cients. It was applied to the 1980 Mt. St. Helensand 79 A.D. Vesuvius eruptions successfully.Because of its 3D nature, the model requiresnumerical simulation in three spatial and onetemporal dimension. This requires considerablecomputing time and limits the user to a smallnumber of possible runs with different parameterseach time. In this study, a modified version (calledHAZMAP) is presented that further developsMacedonio et al. (1988)’s model into a simplifiedsemianalytical 2D model useful for a first approachto reconstruct tephra fall deposit (for a moredetailed com putational description of the HAZMAPmodel seeMacedonio et al., (2004)). We havemodelled the same Plinian falldeposits of theVesuvius 79 A.D. eruption, asMacedonio et al.(1988). The main differences between the twomodels are summarized as follows:
!
The 3D version of the diffusion–advection-settling problem (Eq. (1)) takes account of the verticalcomponent of atmospheric diffusion. In this studyvertical diffusion and the vertical wind component are neglected.
!
Macedonio et al. (1988)used a single parameter-ization for all settling velocity classes to describethe settling velocity dependence with altitude. Inthis study, we adopted a more general ash settlingvelocity model based on the experimental data of Walker et al. (1971)andWilson and Huang(1979).
!
Macedonio et al. (1988)accounted for the tempo-ral variation of the column height (by independent models). We assumed the same eruption columnheight for each phase and an instantaneous massrelease.
!
Macedonio et al. (1988)used a scaled androtated measured summer wind profile estab-lished by statistical observations in southernItaly. In this study, we use several windmodels and test a parameterized wind profile,which is best fitted against the observed fielddata.
!
3D modelling is time intensive to compute,whereas this simplified semianalytical model can perform thousands of runs in few tens of minutes on a PC.In addition, we tested the reliability of the modeland the dependence of the results against the quality/ quantity of the input data.
2. The physical model
Beyond a certain distance from the dynamiceruption column, the dispersion and sedimentationof tephra is governed mainly by wind transport,turbulent diffusion, and the settling of particles bygravity (Armienti et al., 1988; Macedonio et al.,1988). The motion of particles can be described by themass conservation equation as follows:
B
C
j
B
t
þ
W
x
B
C
j
B
x
þ
W
y
B
C
j
B
y
À
B
U
sj
C
j
B
z
¼
K
x
B
2
C
j
B
x
2
þ
K
y
B
2
C
j
B
y
2
þ
K
z
B
2
C
j
B
z
2
þ
S
j
ð
1
Þ
where
C
j
is concentration of particles;
t
the time;
x
,
y
,
z
the spatial coordinates;
W
i
the horizontal windfield (
i
=
x
,
y
);
K
x
and
K
y
are the horizontalatmospheric eddy diffusion coefficients assumedequal so that
K
x
=
K
y
=
K
and
U
s
the settling velo-city; and
S
j
is a source function. Eq. (1) is valid for each class
j
of particles having a given settling ve-locity
U
sj
.In a first-order approach, we assume that thevertical diffusion component is negligible withrespect to the horizontal ones, and that thehorizontal wind components are constant in timeand within the horizontal domain. This assumptionholds for intermediate distances of the order of 100 km or more but becomes less accurate over large distances. Moreover, it is assumed that theeruption column is high enough to neglect terraineffects.Under these assumptions, dividing the verticalcomputational domain into
N
layer
layers on whichsettling and wind velocity is assumed constant, asinMacedonio et al. (2004),the total mass on the
ground is computed as the sum of the contributions
T. Pfeiffer et al. / Journal of Volcanology and Geothermal Research 140 (2005) 273–294
274
from each of the point sources distributed above thevent and from each settling velocity class:
M
G
x
;
y
ð Þ¼
X
N
vs
j
¼
1
X
N
sources
i
¼
1
M
i
f
j
2
pr
2
Gi
exp
Â
"
Àð
x
À
x
Gi
Þ
2
þð
y
À
y
Gi
Þ
2
2
r
2
Gi
#
ð
2
Þ
where
x
Y
Gi
¼
x
Y
0
i
þ
P
k
W
Y
k
D
t
k
and
r
2
Gi
¼
2
K
P
k
D
t
k
are the centre and the variance of the Gaussian,respectively, calculated as inMacedonio et al.(submitted for publication)(
D
t
k
=(
z
k
À
z
k
À
1
)/
v
s
,
k
).Moreover,
N
sources
indicates the source points,
N
vs
the total number of settling velocity classes,
M
i
is the total mass emitted from the point sourcein the layer
i
(
P
i
M
i
¼
M
tot
, with
M
tot
total massinjected into the system) and
f
j
is the fractionof that mass belonging to the settling velocity class
j
(
P
j
f
j
¼
1; for the computational details seeMacedonio et al., (2003)). A summary of thedefinitions of the symbols previously used isreported inTable 1.
3. Parameterizations and input data
3.1. Eruption column
The simplification of the eruption column as alinear source limits the use of the model to areassufficiently far from the vent. The results fromMacedonio et al. (1988)andArmienti et al. (1988)suggest that the critical distance is approximatelygiven by the height of the eruption column itself.Variations of the eruption column with time arethus replaced by a time-averaged column described by a vertical mass distribution function
f
h
(
z
).Describing the vertical mass distribution in aneruption column has been a major goal of a recent research, but in our model, a purely empiricaltreatment is adopted. Modifying theformula sug-
gested bySuzuki (1983)and used inArmienti et al.
(1988)andMacedonio et al. (1988),the vertical
mass concentration is assumed the same for all particle classes as given by:
S x
;
y
;
z
;
t
ð Þ¼
S
0
n
1
À
z H
exp
A z
=
H
À
1
ð Þ½
o
k
Â
d
ð
t
À
t
0
Þ
d
ð
x
À
x
v
Þ
d
ð
y
À
y
v
Þ ð
3
Þ
where
S
(
z
)={1
À
z
/
H
exp [
A
(
z
/
H
À
1)]}
k
is thevertical mass distribution function,
z
the altitude inthe eruption column,
S
0
a normalization factor,
H
the maximum plume height,
A
and
k
are twodimensionless parameters, and
d
is the Dirac’sdistribution (filiform and instantaneous releaseassumption). Eq. (3) is applied as an empiricaldescription of the vertical mass distribution withinthe eruption column with a purely geometricmeaning. The value of the parameter
A
(
b
Suzukicoefficient
Q
) describes the vertical position of themaximum concentration relative to the maximumcolumn height, located at (
A
À
1)/
A
of the max-imum plume height (Fig. 1a). The parameter
k
isa measure of how closely the total mass isconcentrated around the maximum at
H
(
A
À
1)/
A
(Fig. 1 b).Theoretical and empirical observations on buoy-ant plumes (e.g.,Morton et al., 1956; Sparks, 1986)show that the ratio
H
B
/
H
T
between the height of neutral buoyancy of the plume
H
B
and themaximum height
H
T
is usually around 3/4. Wetry to account for this by considering
A
to be about 4 and varying
k
in the range 1–5.
3.1.1. Grain size and settling velocity
Rosin’s law is often used to explain the sizedistribution of erupted pyroclastic material (Suzuki,
Table 1Definitions of the symbols usedSymbol Definition(
W
x
,
W
y
) Horizontal wind field
K
=
K
x
=
K
y
Horizontal diffusion coefficient
U
sj
Settling Velocity of the
j
-th particle class
S
j
Source function for the
j
-th class
M
G
(
x
,
y
) Total mass on the ground at (
x
,
y
)
M
i
Total mass emitted from the
i
-th point source
f
j
Mass fraction of the
j
-th settling velocity class(
x
Gi
,
y
Gi
) Centers of the Gaussian mass distribution
r
Gi
Variances of the Gaussian mass distribution
N
vs
Total number of settling velocity classes
N
sources
Total number of point sources
D
z
k
Thickness of the
k
-th layer
D
t
k
Residence time of particles in the
k
-th layer
T. Pfeiffer et al. / Journal of Volcanology and Geothermal Research 140 (2005) 273–294
275
1983). This probability density function is given by:
f
U
ð Þ¼
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
pr
U
p
exp
À
U
À
l
ð Þ
2
2
r
2
U
#"
ð
4
Þ
where
d
=2
À
U
, mm is the particle diameter,
l
themedian value, and
r
U
the standard deviation of the sizedistribution. Using this parameterization for each typeof component, we are able to describe the settlingvelocity spectrum with only two parameters for eachcomponent (
l
and
r
U
).The bulk of the erupted mass is split into a user-defined number of grain size and classes of components.The settling velocity determines the deposition of all tephra and has a first-order importance for tephradeposition. In any quantitative model of fallout, anaccurate description of the settling velocity of particles is critical.The settling velocity of volcanic particles is acomplex function of particle size, shape, and density.It also depends on the density and viscosity of thesurrounding air. Its value can only be computedapproximately and relies heavily on experimental andempirical data. While for certain regular and industrialshapes a large amount of experimental aerodynamicdata is available, only a few studies have directlymeasured the fall velocities of volcanic particles, themost significant ones being byWalker et al. (1971)andWilson and Huang (1979).Consistent withArmienti et al. (1988), the present model assumes that all particles are alwaystraveling at their terminal settling velocity. This is justified as long as terminal fall velocities are of theorder of a few tens of meters per second, or less.In the case of typical volcanic particles in air, buoyancy forces can be neglected. The terminalsettling velocity of a particle is given by the basicequation when the weight of the particle is balanced by the aerodynamic drag force:
mg
¼
12
C
D
q
a
A
cs
U
2s
ð
5
Þ
where
m
is the mass and
A
cs
a typical cross-sectionalarea of the particle,
C
D
the dimensionless dragcoefficient,
q
a
the density of air,
U
s
the settlingvelocity, and
g
the acceleration due to gravity. Thedrag coefficient itself is a complex function of particleshape and the Reynolds number
Re
=
q
a
dU
s
/
g
a
, whichis a measure of the relative importance of inertialversus viscous forces governing turbulent and laminar flow, respectively (
d
is a
b
characteristic
Q
particlediameter and
g
a
the viscosity of air).At low Reynolds numbers (
Re
b
0.1–1) viscousforces dominate. For spherical particles (Landau andLifchitz, 1971):
C
D
¼
B Re
;
B
¼
24
ð
6
Þ
Fig. 1. Column shape: mass distribution models inside column.
T. Pfeiffer et al. / Journal of Volcanology and Geothermal Research 140 (2005) 273–294
276
This relation is equivalent with Stokes’ law for thesettling velocity:
U
s
¼
q
d
2
g
=
18
g
a
ð
7
Þ
where
q
is particle density.The typical cross-sectional area
A
cs
in Eq. (5)and the characteristic particle diameter
d
in theReynolds number definition are not easy to determinefor irregular shaped bodies. They depend not only onthe shape of the particle but also on the direction of the air flow around it and its turning mode. An exact treatment is impossible, and it is more convenient todefine
d
and
A
cs
as the diameter and cross-sectionof an equivalent sphere, which gives:
A
cs
=
W
d
2
/4.Generally,
d
is not the real mean diameter of the particle defined by the average of the three mainaxes. For subspherical shapes, taking
d
as thediameter of an enclosing sphere or a sphere withthe samevolume as the particlewill be a goodmeasure (Wilson and Huang, 1979). For elongated
particles,
d
and
A
cs
depend very much on theorientation and rotation modes of the particles, andtheir appropriate values might vary widely.Wilson and Huang (1979),who measured theterminal fall velocities of hundreds of small volcanic particles in the range 0.1
b
Re
b
100, showed that it issufficient to define the typical particle diameter as theactual diameter if at least one additional parameter isused to describe fall velocity. They introduced theshape factor
F
=(
a
+
b
)/2
c
, where
a
b
b
b
c
are the three principal diameters of the particle. Then, they showedthat the drag coefficient
C
D
of about 80% of hundredsof investigated tephra particles could be expressed as asimple function of mean diameter
d
, shape function
F
,and Reynolds numbers:
C
D
¼
24
ReF
À
0
:
828
þ
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
:
07
À
F
p ð
8
Þ
Suzuki (1983)proposed a modification to thisformula which appears to be a better fit to bothWilsonand Huang (1979)’s experimental data for submillim-eter clasts and the (less comprehensive) data byWalker et al. (1971)of centimeter-sized pumice andcrystals from Fogo and Askja:
C
D
¼
24
ReF
À
0
:
32
þ
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
:
07
À
F
p ð
9
Þ
Stokes’ law implies that, as long as Reynoldsnumbers are very low, terminal fall velocity doesnot change with air density. This has a significant effect. While larger particles increase their terminalfall velocity rapidly with decreasing air density,i.e., increasing altitude, a small fraction of tephraremains in the regime of low Reynolds numbersand falls much more slowly out of the plume.This effect is greater the higher the eruptioncolumn and could be a major reason for trans- portation to very long distances of tephra in high plumes.For particles falling at high Reynolds numbers(
Re
N
1000), inertial forces are dominant, and the dragcoefficient no longer depends on Reynolds number.Typically, it assumes a constant value for Machnumbers below about 0.7 (Aschenbach, 1972). The
value of
C
D
depends on the shape and orientation of the particle. From data in the literature (e.g,Mironer,1979), drag coefficients for some basic shapes at
Re
N
1000 are:
!
0.44 for spheres
!
1.2 for cylinders
!
0.8–1.05 for cubesVery few experimental data exist for irregular-shaped volcanic particles falling at high Reynoldsnumbers. The experimental studies of Walker et al.(1971), however, suggest that drag coefficients for volcanic particles at
Re
N
1000 are also more similar to those of cylinders than spheres. This is confirmed by an analysis of their experimental data (Fig. 2).
As a result, drag coefficients are expected to bearound
C
D
=1. In the case of intermediate Reynoldsnumbers, the description of the drag coefficient becomes very difficult. Most authors of similar studies have used intermediate analytical expres-sions between the settling laws for low and highReynolds numbers regimes or approximations of experimental data; for instance,Bonadonna et al.(1998)usedKunii and Levenspiel (1969)’s analyt-
ical expressions andMacedonio et al. (submitted for publication)used the formula given byArastoopour et al. (1982).In this study, we have adopted a model based onWalker et al. (1971)andWilson and Huang
T. Pfeiffer et al. / Journal of Volcanology and Geothermal Research 140 (2005) 273–294
277

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