Tephra4D: A Python-Based Model for High-Resolution Tephra Transport and Deposition Simulations—Applications at Sakurajima Volcano, Japan

Abstract

Vulcanian eruptions (short-lived explosions consisting of a rising thermal) occur daily in volcanoes around the world. Such small-scale eruptions represent a challenge in numerical modeling due to local-scale effects, such as the volcano’s topography impact on atmospheric circulation and near-vent plume dynamics, that need to be accounted for. In an effort to improve the applicability of Tephra2, a commonly-used advection-diffusion model, in the case of vulcanian eruptions, a number of key modifications were carried out: (i) the ability to solve the equations over bending plume, (ii) temporally-evolving three-dimensional meteorological fields, (iii) the replacement of the particle diameter distribution with observed particle terminal velocity distribution which provides a simple way to account for the settling velocity variation due to particle shape and density. We verified the advantage of our modified model (Tephra4D) in the tephra dispersion from vulcanian eruptions by comparing the calculations and disdrometer observations of tephra sedimentation from four eruptions at Sakurajima volcano, Japan. The simulations of the eruptions show that Tephra4D is useful for eruptions in which small-scale movement contributes significantly to ash transport mainly due to the consideration for orographic winds in advection.

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atmosphere
Article
Tephra4D: A Python-Based Model for High-Resolution Tephra
Transport and Deposition Simulations—Applications at
Sakurajima Volcano, Japan
Kosei Takishita 1, 2, *, Alexandros P. Poulidis 1,3 and Masato Iguchi 1


Citation: Takishita, K.; Poulidis, A.P.;
Iguchi, M. Tephra4D: A Python-Based
Model for High-Resolution Tephra
Transport and Deposition
Simulations—Applications at
Sakurajima Volcano, Japan.
Atmosphere 2021,12, 331. https://
doi.org/10.3390/atmos12030331
Academic Editor: Young Sunwoo
Received: 31 January 2021
Accepted: 1 March 2021
Published: 4 March 2021
Publisher’s Note: MDPI stays neutral
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Sakurajima Volcano Research Center, DPRI, Kyoto University, Kagoshima 841-1419, Japan;
alepou@uni-bremen.de (A.P.P.); iguchi.masato.8m@kyoto-u.ac.jp (M.I.)
2Division of Earth and Planetary Sciences, Graduate School of Science, Kyoto University,
Kyoto 606-8502, Japan
3Institute of Environmental Physics, University of Bremen, 28359 Bremen, Germany
*Correspondence: takishita.kosei.85s@st.kyoto-u.ac.jp
Abstract:
Vulcanian eruptions (short-lived explosions consisting of a rising thermal) occur daily in
volcanoes around the world. Such small-scale eruptions represent a challenge in numerical modeling
due to local-scale effects, such as the volcano’s topography impact on atmospheric circulation and
near-vent plume dynamics, that need to be accounted for. In an effort to improve the applicability of
Tephra2, a commonly-used advection-diffusion model, in the case of vulcanian eruptions, a number
of key modifications were carried out: (i) the ability to solve the equations over bending plume, (ii)
temporally-evolving three-dimensional meteorological fields, (iii) the replacement of the particle
diameter distribution with observed particle terminal velocity distribution which provides a simple
way to account for the settling velocity variation due to particle shape and density. We verified the
advantage of our modified model (Tephra4D) in the tephra dispersion from vulcanian eruptions
by comparing the calculations and disdrometer observations of tephra sedimentation from four
eruptions at Sakurajima volcano, Japan. The simulations of the eruptions show that Tephra4D is
useful for eruptions in which small-scale movement contributes significantly to ash transport mainly
due to the consideration for orographic winds in advection.
Keywords:
tephra; advection-diffusion model; disdrometer; Sakurajima; Tephra2; Tephra4D;
Parsivel; WRF
1. Introduction
Volcanic eruptions release tephra to the atmosphere causing a hazard that can ad-
versely affect the lives and livelihood of surrounding populations, leading to damages in
sectors such as agriculture, forestry, fisheries. Furthermore, airborne tephra can be widely
dispersed and have a serious impact on aircraft operations, as shown in the 2010 eruption
of Eyjafjallajökull volcano [
1
]. For these hazard assessments, numerical simulations using
advection-diffusion models are used to predict the dispersion of tephra in the atmosphere
and the mass of tephra deposits [2].
After an eruption and the formation of a plume, tephra is advected by the wind,
diffused due to turbulent processes, and descends due to its mass. In order to simulate the
transport and deposition of tephra, numerical models require input parameters such as
the plume height, total mass of tephra, wind field, and particle size, all of which are either
based on observations, estimated, or simulated. Based on the meteorological conditions
and the size of the eruptions, tephra transport and deposition can be simulated on scales of
tens to hundreds of kilometers (e.g., [3–5]).
Tephra deposition prediction models that can reproduce small-scale eruptions are
necessary for the study of volcanic plume dynamics. Up until now, most studies have
Atmosphere 2021,12, 331. https://doi.org/10.3390/atmos12030331 https://www.mdpi.com/journal/atmosphere

Atmosphere 2021,12, 331 2 of 21
focused on large-scale eruptions such as Plinian eruptions, but they have not been able
to establish parameters such as the distribution of the segregation height of tephra in the
plume (tephra segregation profile (TSP); e.g., [
5
,
6
]) and the interaction between particles
and between particles and fluid (e.g., [
7
,
8
]). This is partly because large eruptions are
infrequent, leading to the shortage of usable observations. Vulcanian eruptions present
an opportunity to cover this lack of data due to their high frequency and relative safety of
data acquisition.
For tephra transport numerical modeling to be suitable for vulcanian eruptions, it
is first necessary to improve the consideration of the advection process. When tephra
segregates from the plume at a low altitude from the surface of a volcano, it is important
to take into account the bending of the ascending plume [
9
] and the heterogeneity of the
wind field caused by the presence of the mountain (e.g., [10,11]).
When initializing a tephra transport simulation, it is common to assign grain size dis-
tribution to characterize the range of released particles. However, laboratory experiments
have shown that the settling velocity of a particle of the same size depends on its effective
density, shape, and whether or not it is bound to other particles (such as [
6
,
12
]), suggesting
the importance of constraining the descent process not by the particle size distribution but
by the settling velocity distribution. Due to the limitation of the observation, the particle
size distribution has been assigned as the input parameter of the previous tephra deposit
prediction model, and the settling velocity distribution was approximated from the particle
size distribution based on the simple parameterizations. Particles with irregular effective
density were calculated with an option (e.g., [
13
]) or neglected in these models. If the
model handles a settling velocity distribution as an input parameter, such considerations
can be made by simply changing the mass distribution without any special options.
In recent years, optical disdrometers have been increasingly used to observe tephra
sedimentation [
8
,
14
,
15
] because they can measure the settling velocity of falling particles
with a high temporal resolution. This allows for an alternative strategy: the characterization
of the released tephra through a settling velocity distribution.
As such, in this study, we develop an advection-diffusion model Tephra4D based on
the advection-diffusion model Tephra2 [
3
]. Tephra4D can take into account the spatial
distribution of the time series of three-component wind velocity field and atmospheric
density in the advection process and constrain the descent process by the settling velocity.
Using disdrometer data from four eruptions from Sakurajima volcano, Japan, two sets
of simulations are carried out. A simulation with the simple wind field is performed
with a reimplemented model of Tephra2 constrained by the settling velocity distribution
and simulations using the modified Tephra4D model. By comparing the results with the
observations using a disdrometer network, we examine how the specification changes
made in Tephra4D contribute to the improvement of the calculation and which points need
further improvement.
The paper is structured as follows. Initially, the detailed calculation of the tephra
deposit by Tephra4D is described in Section 2. Then the eruptions studied, model setup,
and the results of the calculation are detailed in Section 3. In Section 4, we discuss the
advantages of Tephra4D and important factors. Finally, Section 5summarizes the main
conclusions of the study.
2. Model Description
2.1. The Advection-Diffusion Model Tephra2 and Its Modifications
Tephra2 is an advection-diffusion model improved from the analytical model of [
6
],
which numerically solves the equation of terminal velocity and dispersion [
16
]. The
migration path of the center of gravity of the particle cluster as they segregate from the
plume and fall into the atmosphere (hereinafter referred to as the “trajectory”) is calculated
and then the tephra deposit load is obtained considering the diffusion process (Figure 1).

Atmosphere 2021,12, 331 3 of 21
Figure 1.
Schematic representation of the tephra deposit at site A in Tephra2. The solid red line
indicates the movement of the center of gravity of the particles segregated from the plume, and the
dotted red line indicates the range of distance 3
σ
from the trajectory. The symbols in the figure are
defined following Equations (1)–(3).
Tephra2 employs a horizontally homogeneous steady state, does not account for
plume bending, and particles are classified by diameter. These aspects of the model need to
be improved in order to successfully model vulcanian eruptions. We do this by employing
temporally evolving wind fields obtained from the Weather and Research and Forecasting
(WRF) model, including a mechanism for plume bending, and classifying the particles by
terminal velocities instead of diameter. We refer to this improved model as Tephra4D.
A model that implements plume bending in Tephra2 was developed by [
9
], while
in this study we implement another simple assumption. The vertical velocity when the
plume rises is calculated in the same way as the particle falls. The ascending velocity and
the horizontal coordinate of the segregation point are independent of the settling velocity.
Tephra4D takes into account wind heterogeneity in advection, but not in diffusion,
and follows the assumption in Tephra2 that particles diffuse concentrically only in the
horizontal direction. This is because the horizontal size of the plume is about 1.5 km, even
for the largest plume in this study, and the horizontal wind is approximately uniform
within this region. We reimplemented Tephra2 to use arbitrary TSP and settling velocity
distributions as input parameters for control experiments. This reimplemented version
is referred to as Tephra2
PY
in this study. The difference of specifications among Tephra2,
Tephra2PY, Tephra4D are described in Table 1.
Table 1. The specifications among three models.
Parameters Tephra2 Tephra2PY Tephra4D
∆hx(km) - - ≈0.3
∆t(min) - - 10
Continuity of σat
t = FTT - Equation (4) Equation (4)
Plume bending - - Equations (7) and (8)
vtdistribution Calculated from dDirectly substituted Directly substituted
Wind below vent Equation (10) Equation (10) Same as above vent
(WRF)
ρa(h) Equation (11) Equation (11) WRF
Settling velocity [17] [17] Equation (15)
TSP Equation (20) Equations (17)–(20) Equations (17)–(20)
Language C Python3 Python3

Atmosphere 2021,12, 331 4 of 21
We prepare a representative particle corresponding to each settling velocity class.
The movement of the center of gravity of a group of particles is calculated as the move-
ment of the representative particle. The effective density of the particle is assumed to be
2640 kg/m
3
[
18
]. The diameter and shape parameter of the particle are adjusted to take
the same settling velocity as the median of the settling velocity class of the disdrometer at
0 m asl.
Let L
A
(h
seg
,d) be the areal concentration of particles of settling velocity v
t
arriving at
point A on the ground from a plume at the segregation height h
seg
, then the tephra deposit
load at point A S
A
can be expressed as the sum of the areal concentrations of particles as a
function of the segregation height and terminal velocity:
SA=∑
vt
∑
hseg
ρAhseg ,vt(1)
Assuming that a falling particle cluster diffuses only horizontally and that the diffusion
equation is expressed by a two-dimensional Gaussian distribution (variance
σ
) of the
horizontal deviation (x,y) from the center of gravity of the particle cluster, the areal
concentration L
A
(h
seg
,v
t
) at point A, (x
A
,y
A
) away from the center when the center reaches
an altitude of point A, is obtained from the following equation:
LAhseg ,vt=Mhseg ,vt
πσAhseg,vt2exp −x2
A+y2
A
σAhseg ,vt2!(2)
where Mis the mass of the particle cluster with settling velocity v
t
segregated from the
altitude h
seg
and
σA
is the dispersion of the cluster at the altitude at which point A is located.
Dispersion
σ2
is expressed as a function of dispersion time tand empirically given by an
expression proportional to the first power of twhen tdoes not exceed a threshold FTT (Fall
Time Threshold) or an expression proportional to the 2.5 power of twhen texceeds FTT [
6
].
In this study, it is assumed that the horizontal spread of a plume is also caused by the
diffusion of pyroclastic particles when the plume rises blowing in the wind. Assuming t=
tA(the time between the departure at the vent and the arrival at point A), σ2is calculated
as follows:
σAhseg ,vt2=4KtA,|tA<FTT
1.6CtA2.5,tA≥FTT (3)
where Kand Care the diffusion coefficient. FTT is defined to be 3600 s, the setting value in
Tephra2, and considering the continuity of
σ
at t=FTT, which is not considered in Tephra2
and considered in Tephra2PY,Cis defined as follows:
C=2.5K
FTT1.5 (4)
The relationship between the radius of the plume rand the elevation from the vent h
is empirically given as follows [19]:
r=0.34h(5)
and in Tephra2 and Tephra2PY the following relationship is also given [3]:
r=3σ(6)
This relationship is also followed in Tephra4D. Using Equations (3)–(6), the time
between the departure at the vent and the segregation at altitude h
seg
,t
up
(h
seg
), is obtained
as follows:
tup hseg=




3.2×10−3h2
seg
K,|tup <FTT
8×10−3h2
seg
C0.4
,|tup ≥FTT (7)

Atmosphere 2021,12, 331 5 of 21
Applying Equations (3), (5) and (6) to a vulcanian eruption at Sakurajima, Kwas
estimated from the time evolution of the plume top height. The height was read from
the live camera image [
20
] every minute as shown in Figure 2. The optimal value is
K= 100 m2/s
although there is a large variation. The vertical velocity when particles rise
through the plume is calculated based on Equation (7). We do not take into account the
temporal variation of the mass eruption rate.
Figure 2.
Temporal changes in plume height (H
p
) during eruptions from January to November 2019.
Lines are based on theoretical values.
2.2. The Trajectory Calculation
In the trajectory calculation, we consider a Cartesian coordinate system with the east-
ward, northward, and upward as the positive directions of x-, y-, and z-axis, respectively.
Since the position of the meteorological field data is based on the UTM coordinate system in
the horizontal direction and the altitude asl in the vertical direction, as shown in
Figure 3
,
the particle exists in a hexahedron. The meteorological field inside this hexahedron is
assumed to be a spatially homogeneous field represented by the value given to point C in
Table 2. The entire meteorological field will be composed of a large number of hexahedra,
each with a homogeneous wind field and atmospheric density.
Figure 3.
The movement of particles inside a hexahedron with eight points at the vertices given a
weather field W.

Atmosphere 2021,12, 331 6 of 21
Table 2. The symbols to be used in trajectory calculation. The index nindicates nth step.
Symbol Definition
z=z0,z1The lower and the higher surface of the
hexahedron.
A(x10,y10 ,z0), B(x11,y11 ,z0),C(x00,y00 ,z0),
D(x01,y01 ,z0)
The coordinate of vertices of the lower surface.
(xn,yn,zn)The coordinate of a particle inside a
hexahedron.
(un,vn,wn) The wind vector in the hexahedron.
vtn The terminal velocity of the particle.
tnTime to start.
∆tx,∆ty,∆tzTime for the particle to reach the interface
almost vertical to the x-, y-, and z-axis.
In each step, after the particle starts to move, the particle goes straight in the hexahe-
dron, reaches the interface of the hexahedron, and moves to the next step in the neighboring
hexahedron. The movement of the particle at nth step is expressed as the following recur-
sion formula:


xn+1
yn+1
zn+1
=

xn
yn
zn
+min∆tx,∆ty,∆tz

un
vn
wn+vt,n
(8)
Each symbol is described in Table 2. The recursion formula about time is as follows:
tn+1=tn+min∆tx,∆ty,∆tz(9)
The same calculation is performed inside the hexahedron which the particle entered.
If the particle moves to a neighboring hexahedron and then immediately returns to the
original hexahedron, i.e., the hexahedron on which the particle resides at a given step is the
same as the two steps earlier, the coordinate shift is calculated with the velocity component
in the direction of boundary travel set to zero and the other velocity components set to
the average of the velocity fields on the two hexahedra. Such a sequential calculation of
trajectory is repeated until the particle reaches the ground or the horizontal limit of the
calculation domain every indefinite spatial step. The calculation of
∆
t
x
,
∆
t
y
,
∆
t
z
is detailed
in Supplementary Material S1.
2.3. Wind Field and Atmospheric Density
Tephra4D uses temporally evolving three-component wind and atmospheric density
data. The data were computed using the WRF model [
21
] with a horizontal grid spacing of
300 m, 58 layers in the vertical, and a temporal output of 10 min. The results from WRF are
interpolated to a vertical grid to be used by Tephra4D. The detailed calculation of the wind
field by WRF is described in Appendix C.
In Tephra2 and Tephra2
PY
, the wind field is assumed to be horizontally uniform and
does not have a vertical component. The wind field above the crater is interpolated from
the wind speed and direction given by the user for each section. The wind speed at sea
level is assumed to be zero and the wind speed v(h) at an altitude hasl under the crater is
assumed as follows:
v(h)=h
hvent
v(hvent)(10)

Atmosphere 2021,12, 331 7 of 21
where h
vent
is the altitude of the vent asl. Assuming the density of fluid at 0 m asl as
ρa0
,
the density of fluid ρaat the altitude his calculated following the hypsometric formula:
ρa(h)=ρa0 exp−h
8200 (11)
where the unit of h and 8200 is m.
The simulated wind field for the eruption on 16 June 2018 is shown as an illustrative
example. Tephra transport occurred to the west of the vent, agreeing with the resolved
wind direction (Figure 4a). The atmospheric flow over the volcano is dominated by
orographic waves, indicated by the oscillation seen in the west side (leeside) of the volcano
(repeated pattern of downwards and upwards wind extending outwards) [
22
]. Such
orographic activity has been shown to play an important role in the transport of tephra
over complex topography [10].
Figure 4.
(
a
) Spatial distribution of the wind velocity vector (horizontal component) at 1000 m asl,
(
b
) vertical velocity distribution at 1000 m asl, and (
c
) wind velocity field in east–west section through
the Minamidake crater at the onset of the eruption on 16 June 2018. Positive values indicating
upwards motion in (
b
). Shading indicates vertical velocity (blue for negative and red for positive
values) in (
b
). Solid gray lines in (
c
) are topographic features used for trajectory calculations and
dashed gray lines are topographic features used for calculating the tephra deposit load distribution.
The vertical direction is emphasized twice as much as the horizontal direction in both elevation
and wind.
2.4. The Settling Velocity of Pyroclastic Particles
We consider that the difference in settling velocity for the same diameter corresponds
to the difference in effective density due to the shape of the particles and the degree
of agglomeration. We first obtain the relationship among the effective density of the
pyroclastic particle
ρp
(kg/m
3
), the terminal velocity v
t
(m/s), and particle diameter d
(mm). Assuming that the shape of the pyroclastic particle is a spheroid, the relationship

Atmosphere 2021,12, 331 8 of 21
between
ρp
,v
t
and dis represented by the following set of equations connected by the drag
coefficient CDand the Reynolds number Ra[6]:
vt=s4gdρp
3CDρa(12)
CD=24
Ra
F−0.32 +2√1.07 −F(13)
Ra=ρavtd
ηa(14)
where gis the gravitational acceleration,
ηa
,
ρa
are the viscosity and density of the sur-
rounding fluid (i.e., the atmosphere in this study), and Fis the shape parameter of the
particles. Substituting Equations (13) and (14) into Equation (12) we get:
vt=ρdgd2
9ηaF−0.32 +q81η2
aF−0.64 +3
2ρaρpgd3√1.07 −F
(15)
Based on the diagram of the diameter and shape parameter Fof the pyroclastic
particles at Stromboli volcano [23], Fis calculated using the following equation:
F=0.81 +0.03 log2(d/1000)(16)
Assuming that v
t
and dare equal to the settling velocity and particle size observed by the
disdrometer, respectively,
ηa
,
ρa
are the values for the atmosphere at 20
◦
C,
1.8 ×10−5Pa·s
,
1.205 kg/m
3
, the settling velocity, effective particle density is calculated as shown
in Figure 5
.
Figure 5.
Effective density distribution calculated by Equation (15). The black dotted line indicates
0.1. Unit is kg/m3.
Tephra2 and Tephra2
PY
calculates the terminal velocity from [
17
], while Tephra4D
calculates it from Equation (15), which is used to calculate the observed tephra deposit load
in Appendix B. As shown in Supplementary Material S2, the distance to reach the terminal
velocity is negligible concerning the height of the plume top, so the particles are considered
to have reached their terminal velocity when they are segregated from the plume. The
spatial distribution of the terminal velocity of a pyroclastic particle with a terminal velocity
of 2.2 m/s on the ground is shown in Figure 6a for the eruption on
16 June 2018
. The
terminal velocity decreased by about 20% between 5 and 0 km asl. Due to the influence
of downward orographic wind to 3 km asl (Figure 6b), at the point where the downwind
is strongest, the particle falls with 8.0 m/s in the settling velocity, which is the sum of
the terminal velocity of 2.3 m/s plus the downward wind of 5.7 m/s. Thus, when strong
mountain waves are formed, the change in settling velocity is more strongly affected by
the change in the downward wind than by the change in atmospheric density.

Atmosphere 2021,12, 331 9 of 21
Figure 6.
Spatial velocity distribution of pyroclastic particles with a terminal velocity of 2.2 m/s at
the ground in the east–west section through the Minamidake crater at the onset of the eruption on
16 June 2018
. (
a
) Particle terminal velocity and (
b
) the sum of terminal velocity and vertical wind.
Solid gray lines are topography used to calculate trajectories, and dashed gray lines are topography
used to calculate tephra deposit distribution. The vertical direction is emphasized twice as much as
the horizontal direction. Shading indicates the total settling velocity of a particle (red upwards and
blue downwards motion) in (b).
2.5. Tephra Segregation Profile
Since TSP M(h
seg
,v
t
), which is substituted in Equation (2), is highly arbitrary, it was
constrained using the model of the distribution function. At an altitude interval of
∆
h,
the mass of tephra with terminal velocity v
t
at altitude h
seg
discretized between the vent
altitude h
vent
to the plume altitude h
vent
+h
plume
is given by the following equation defining
the function of TSP as m(h):
Mhseg ,vt=Mtotal (vt)Rhseg +∆h
hseg m(h)dh
R+∞
−∞m(h)dh (17)
where M
total
(v
t
) is the settling velocity distribution of the total mass fraction of tephra. In
this study, the observed settling velocity distribution in each eruption is applied to M(v
t
).
We applied three distribution functions m(h) in this study.
1. Logarithmic Gauss distribution with top concentration;
m(h)=


1
√2π(hvent+0.9 hplume −h)exp−1
2log hvent+0.9 hplume −h
0.1 hplume 2,|h<hvent +0.9 hplume
0, |h≥hvent +0.9 hplume
(18)
2. Logarithmic Gauss distribution with bottom concentration;
m(h)=1
√2π(h−hvent)exp
−1
2 log h−hvent
0.1 hplume !2
(19)
3. Uniform distribution.
m(h)=1, |hvent +hplume ≥h≥hvent
0, |h>hvent +hplume ∨hvent ≥h(20)
These distributions are shown in Figure 7. TSP 1 and 2 have m(h) > 0 below the vent
and above the plume top, respectively, and h
seg
is restricted to the height from the vent to
the plume top. As such, the sum of Min Equation (17) is smaller than Mtotal .

Atmosphere 2021,12, 331 10 of 21
Figure 7. TSP of three distributions. The numbers in the figure represent the case numbers.
3. Benchmarking
3.1. The Eruptions Studied
In this study, we used eruptions from Sakurajima volcano, one of the most active
and closely monitored volcanoes in Japan [
24
] to evaluate Tephra2
PY
and Tephra4D. The
eruptive activity of Sakurajima volcano is detailed in Appendix A. We analyzed four
eruptions shown in Table 3that occurred at the Minamidake crater, one large eruption (L),
two moderate eruptions (M1, M2), and one small eruption (S). A disdrometer network was
installed on the island of Sakurajima to observe the tephra deposit load for error evaluation.
The detailed observation is described in Appendix B. The height and direction of the plume
tops are announced by JMA (eruptions L and M1), and in case the plume was obscured by
clouds, they were estimated using the X-band MP radar image (eruptions M2 and S) [
24
,
25
].
Eruption volumes were calculated based on [
26
] from the strain changes associated with
the eruption at Arimura (AVOT in Figure A4). The range of the plume height covers the
typical height in Sakurajima (Figure A2a). The eruptions were chosen as they represent
a range of typical eruptions from Sakurajima that led to tephra sedimentation over the
volcano at a large number of disdrometers (≥4).
Table 3. Eruptions used as case studies.
id Eruption
Start
Plume
Height
(m agl)
Direction Stations Detected
Sites Ejecta (t)
L2018/6/16
7:19 4700 W 12 4 22,000
M1 2018/5/29
3:34 2500 T 11 5 8100
M2 2018/6/9
21:27 2500 T 12 6 10,100
S
2018/11/27
9:01 1500 NE 12 5 3200
In the two eruptions where the direction of the flow was above the crater (T in the table), tephra deposit was
detected at stations from north to west as seen from the crater.
The temporal variation of the tephra deposit load at six sites associated with this
eruption is shown in Figure 8a. Since vulcanian eruptions and subsequent continuous
tephra emissions have different parameters such as plume height and TSP, we extract only
the mass of pyroclastic particles ejected immediately after the eruption. To do this we
consider that sedimentation from the initial explosion ends if there is a 10-min period of
nondetection for each disdrometer location. For the analyzed sample, the settling velocity
distributions of the tephra deposit load for the four eruptions were calculated according
to Appendix B(Figure 8b) and applied to M
total
(v
t
)in Equation (17). The detected settling
velocity range was widest and the heaviest tephra deposit was detected during eruption

Atmosphere 2021,12, 331 11 of 21
M2. The spatial distributions are shown in Figure 9. The azimuths of tephra deposit were
distributed with the range of about 90
◦
in all the eruptions. Eruptions L to S produced
the largest tephra deposit in the west, northwest, west-northwest, and north, respectively.
More than 1 kg/m
2
of tephra deposit was detected at a total of three sites in eruptions L
and M2.
Figure 8.
(
a
) The time series of the tephra deposit load in eruption M2 showing deposit load
for particles with all the diameter on the six stations detected. (
b
) The observed settling velocity
distribution in each eruption. The legend corresponds to the eruption symbols in Table 3.
Figure 9. Tephra deposition load distribution for eruption: (a) L, (b) M1, (c) M2, (d) S.
3.2. Model Setup
In this study, the settings in Table 4were applied.
∆
his the vertical resolution of the
wind field and the slice of plume model, h
vent
is the altitude of the vent,
∆
tis the temporal
resolution of the wind field,
ρa
(h) is the atmospheric density at the altitude h,Kand Care
the diffusion coefficients in Equation (3), and ηais atmospheric viscosity.

Atmosphere 2021,12, 331 12 of 21
Table 4. Model parameters applied in the simulations.
Parameters Tephra2PY Tephra4D
∆hz(km) 0.1 0.1
hvent (m asl) 1000 1000
ρa(0) (kg/m3)1.205 1.205
K, C (m2/s, m2/s2.5) 100, 1.2 ×10−3100, 1.2 ×10−3
FTT (s) 3600 3600
ηa(Pa s) 1.8 ×10−51.8 ×10−5
3.3. Results Comparison
The trajectories of tephra calculated in Tephra2
PY
and Tephra4D are shown in
Figure 10
.
The horizontal position of the segregation point at the top of the plume was calculated
almost the same as the horizontal coordinate of the vent in eruptions M1, M2, S, and in the
same direction where the deposit was detected in eruption L. The maximum horizontal
travel distance during ascent was calculated to be about 1.5 km, and the consideration of
plume bending in this study does not seem to have a significant impact. Trajectories in
Tephra4D change direction more frequently than those in Tephra2
PY
, reflecting the con-
sideration of horizontal heterogeneity in wind fields. The point reached on the ground is
generally nearer in Tephra4D than in Tephra2
PY
when particles segregate from low altitude
and farther when particles segregate from high altitude. This is because the downward
wind blows near the ground and the upward wind blows far from the ground in the region
just leeward from the vent in the cases in this study (Figure 4c).
Figure 10.
The trajectories of tephra calculated in Tephra4D (solid lines) and the reimplemented
model of Tephra2
PY
(dashed lines) with settling velocity (
a–d
) 1.1, (
e–h
) 2.2, (
i–l
) 4.4 m/s. Subplots
in the same columns correspond to the same eruption labeled on the top and subplots in the same
rows correspond to the same settling velocity labeled on the left. Colors indicate the segregation
height and black solid lines are trajectories when tephra rises.
Based on these trajectories, the tephra deposit load distributions are calculated as
shown in Figure 11. Overall the change in the way the wind field is used in the cal-

Atmosphere 2021,12, 331 13 of 21
culations can have a considerable impact. As with the trajectories, the most significant
impact is seen close to the vent as enhanced deposition leads to larger deposit loads in the
Tephra4D calculations.
Figure 11.
The tephra deposit load distribution calculated by (
a
–
e
) Tephra2
PY
and (
f
–
j
) Tepra4D. Subplots in the same
columns correspond to the same eruption and settling velocity labeled on the top and subplots in the same rows correspond
to the same model labeled on the left.
We compared the calculation accuracy of Tephra2
PY
and Tephra4D by the root mean
square error (RMSE) and mean absolute percent error (MAPE) between calculated tephra
deposit load S
cal
and observed tephra deposit load by disdrometer S
obs
. RMSE and MAPE
are calculated using the following equations, respectively:
RMSE(er,vt,TSP)=r1
N∑N
n=1(Scal (er,vt,TSP,n)−Sobs (er,vt,TSP,n))2(21)
MAPE(er,vt,TSP)=1
N∑N
n=1
Scal (er,vt,TSP,n)−Sobs (er,vt,TSP,n)
Sobs (er,vt,TSP,n)(22)
where er,nare eruption and observation site, respectively. RMSE is shown in units of
kg/m
2
while MAPE at a percentage. The sites where S
obs
or S
cal
is bigger than 0 are
considered in Equation (21) and the sites where S
obs
is bigger than 0 are considered in
Equation (22). Nis the number of considered sites. RMSE and MAPE are classified by the
settling velocity class. The number of combinations of eruption, TSP, and settling velocity
class where the number of sites where S
cal
> 0 and S
obs
> 0 N’, are shown in Table 5. Based
on the disdrometer observations, calculations using Tephra4D consistently improve results
for particles with settling velocities less than 0.8 m/s but accuracy significantly drops for
larger particles, particularly for settling velocities over 7.2 m/s.
Table 5. The number of combinations of eruption, TSP, and settling velocity class.
0–0.8 m/s 0.8–2.4 m/s 2.4–7.2 m/s 7.2–22.4 m/s
N> 0 78 96 75 15
N’>0
(Tephra2PY)16 96 59 12
N’>0
(Tephra4D) 58 96 43 3
The settling velocity groups where more than half of the combinations where N’ > 0
are 0.8–2.4 and 2.4–7.2 m/s. RMSE and MAPE distributions in these groups are shown in
Figure 12. The ranges of RMSE are generally similar with a small but notable reduction
in Tephra4D, while MAPE is reduced in Tephra4D especially with the settling velocity
of 2.4–7.2 m/s. RMSE and MAPE distribution for the individual eruption is shown in
Figure 13. Both RMSE and MAPE distributions improved significantly for the Tephra4D

Atmosphere 2021,12, 331 14 of 21
simulation in eruption L. In eruption M1, both the maximum and minimum of RMSE and
MAPE reduced but the median increased with settling velocity 0.8–2.4 m/s. Accuracy
was decreased based on both RMSE and MAPE and for both settling velocity groups in
eruption M2. In eruption S, the maximum of RMSE and MAPE decreased but the minimum
increased with settling velocity of 0.8–2.4 m/s.
Figure 12.
(
a
) RMSE and (
b
) MAPE distributions classified by settling velocity. Box-and-whisker
diagrams are based on the median (orange line in the results of Tephra2PY and blue line in the results
of Tephra4D), the 25th–75th percentile (boxes), and the range of 1.5 times the box (whiskers), with
outliers circled.
Figure 13.
(
a
–
d
) RMSE and (
e
–
h
) MAPE distributions for the results of Tephra2
PY
and Tephra4D
classified by settling velocity and eruption. Subplots in the same columns correspond to the same
eruption labeled on the top and subplots in the same rows correspond to the index on the left.
Box-and-whisker diagrams are based as shown in Figure 12.
4. Discussion
We estimated tephra deposits in four eruptions with various plume heights and wind
fields in Tephra4D and Tephra2
PY
. The tephra deposit load calculation is improved by
Tephra4D in eruptions L and S compared with Tephra2
PY
, were mostly unaffected in
eruption M1, and worsened in eruption M2. The order of plume height is L, M1, M2,
and S (M1 and M2 are the same), and there is no correlation between plume height and

Atmosphere 2021,12, 331 15 of 21
Tephra4D’s advantage. Since L was the eruption with the most dynamically moving
trajectories during ascent and descent, and S was the eruption with the smallest plume
height, Tephra4D is suggested to be useful for eruptions in which small-scale movement
contributes significantly to ash transport.
The most significant reason for this advantage of Tephra4D is suggested to be the
orographic wind consideration in the advection process, whose importance in tephra
dispersion has been suggested in [
10
,
27
]. As shown in Figure 4b, strong downslope winds
are present within 3 km of the crater, and the tephra deposit load near the crater was
heavier in Tephra4D with vertical winds than in Tephra2
PY
without vertical winds. Such
difference in consideration into orographic winds affects the settling velocity range of
tephra to deposit in the calculation. The settling velocity range where N’>0 is shown in
Table 6. The lower limits of the velocity were more accurate in Tephra4D, while the upper
limits of the settling velocity were more accurate in Tephra2
PY
, suggesting that Tephra4D
is useful for eruptions in which many particles with low settling velocity are ejected. The
wind in the cases M1 and M2 was relatively weaker than that in the cases L and S leading
to weaker orographic wave activity (Supplementary material S3). This can explain the lack
of improvement for these two cases.
Table 6.
The settling velocity range of the observed and estimated to be deposited at at least one
observation site. Unit is m/s.
Observation/Model L M1 M2 S
Disdrometer 0.15–7.6 0.25–4.4 0.05–12 0.35–3.8
Tephra2PY 0.85–7.6 0.65–3 0.45–12 0.75–3
Tephra4D 0.35–7.6 0.45–2.6 0.15–4.4 0.35–2.6
Despite the reproducibility of orographic winds in Tephra4D, Tephra2PY sometimes
produces calculations more consistent with observations. This may be because Tephra4D
does not take into account the orographic flows in the diffusion process. The movement of
the particle cluster is represented by a single trajectory for each settling velocity and segre-
gation height, and the diffusion of the particle cluster is considered to be concentric without
considering the heterogeneity of the wind field. The wind heterogeneity in the cases L and
S might be strong enough that the advantage of considering it in advection outweighed
the disadvantage of not considering it in diffusion. Implementing the effect of orographic
wind on diffusion, Tephra4D will contribute to the estimation of unknown parameters
such as TSP and the interaction between particles and between particles and fluid.
The accuracy of the calculation of tephra dispersion affects the diffusion coefficient.
When the diffusion coefficient is small, the accuracy of the calculation of the tephra deposit
distribution is greatly affected by the accuracy of the particle movement estimation, and it is
difficult to perform a highly accurate calculation in substituting a small diffusion coefficient.
In this study, we applied 100 m
2
/s as Kfrom the time evolution of the plume top height
and the tangential width of tephra deposit are properly reproduced. In previous studies, K
was estimated as shown in Table 7and is longer than Kin the present study. In the studies
for the eruptions from Ruapehu [
28
] and Etna [
29
], diffusion during the plume rising was
neglected and Kon descent might be overestimated, but Kin Tephra4D was small even
taking it into account. Since these previous studies gave a horizontally homogeneous wind
field, the diffusion coefficient likely compensated for the three-dimensional heterogeneity
of the wind field. In Tephra4D, the heterogeneity in advection and the mountain wind
contributed to the decrease in the diffusion coefficient compared to previous studies.
By considering the heterogeneity in the diffusion as well as in the advection, we might
calculate more accurate dispersion and reproduce the local variation of the tephra deposit
load distribution more accurately.

Atmosphere 2021,12, 331 16 of 21
Table 7. Kapplied in the previous studies.
Eruption Plume Height (km) K(m2/s) Reference
Izu-Oshima, 1983 10–12 500 [5]
Ruapehu, 1996 8.5 1100 [28]
Etna, 2001 3.5–5 2000 [29]
5. Conclusions
The study of tephra dispersion from small-scale eruptions is required because of its
high frequency. We created a new advection-diffusion model modified from Tephra2
PY
in order to study small-scale eruptions, which need to consider local deviations of wind
fields. Two advection-diffusion models with different wind fields and ascending plume
trajectories were used to calculate the tephra deposit load distribution for the vulcanian
eruptions from Sakurajima volcano, Japan.
Results from Tephra4D, the model developed in this study, show a shift in the de-
position of tephra closer to the vent compared to the conventional model and enhanced
lateral diffusion of the tephra cloud leading to wider deposition patterns. The model is
expected to be useful for eruptions in which small-scale movement contributes significantly
to ash transport. The main factor that led to these differences was the consideration of
orographic winds. The comparison against results using Tephra2
PY
shows reduced errors
by considering the heterogeneity of the wind field, potentially leading to improvement in
our study to establish unknown parameters in plume dynamics.
Supplementary Materials:
The following are available online at https://www.mdpi.com/2073-443
3/12/3/331/s1, S1. The calculation of the elapsed time to reach the boundary of a hexahedron in
trajectory tracking, S2. Vertical travel distance from segregation to the achievement of the terminal
velocity, S3. Wind field in the cases of current study
Author Contributions:
Conceptualization, K.T., A.P.P., and M.I.; methodology, K.T., A.P.P., and M.I.;
software, K.T.; validation, K.T.; formal analysis, K.T.; investigation, K.T., and M.I.; resources, A.P.P.
and M.I.; data curation, K.T.; writing—original draft preparation, K.T.; writing—review and editing,
K.T., A.P.P. and M.I.; visualization, K.T.; supervision, M.I.; project administration, M.I.; funding
acquisition, M.I. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by Integrated Program for the Next Generation Volcano Research
and Human Resource Management project, funded by the Japan Ministry of Education, Culture,
Sports, Science and Technology (MEXT).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data presented in this study are available on request from the
corresponding author. The source code for the models used is available at the following GitHub
repository: https://github.com/Kosei-Takishita/Tephra2_PY (accessed on 1 March 2021) (Tephra2
PY
)
and https://github.com/Kosei-Takishita/Tephra4D (accessed on 1 March 2021) (Tephra4D).
Acknowledgments:
All simulations were carried out on the Kyoto University supercomputer system.
The authors would like to thank two anonymous reviewers for their supportive comments that helped
improved the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A Sakurajima
The study focuses on Sakurajima volcano (Figure A1a,b), one of the most active and
closely monitored volcanoes in Japan [24]. Eruptive activity has occurred continuously at
the Minamidake crater (Figure A1c) since 1955 with varying levels of activity. The volcano
has one dormant crater (Kitadake) and two active craters, Minamidake and the Showa
crater. The latter is a parasitic crater formed in 1939 and bore the main activity between

Atmosphere 2021,12, 331 17 of 21
2006 and 2016 [
11
]. Minamidake is further divided into two nearby craters (A and B) and
has been bearing the main activity since October 2017.
Figure A1.
(
a
) The location of Sakurajima (red triangle) and large cities within Kagoshima pref.
(
b
) The general view of Sakurajima, and (
c
) the locations of the Minamidake A and B craters and the
Showa crater around the summit of Minamidake. Both axes of (
b
) are based on the Minamidake
A crater.
The majority of explosive activity from the volcano since 1955 has been ash-rich
vulcanian eruptions, occurring after increasing pressure causes the brittle plug over the
vent to destabilize [
30
]. The duration of such eruptions is commonly limited to a few
minutes, up to an hour [
11
,
31
]. Plumes from Sakurajima can reach up to 10,500 m above
ground level (agl) in height [
32
]; however, plume heights are commonly constrained within
500–5000 m [
11
] with the number of eruptions decreasing exponentially against plume
height (Figure A2a).
Figure A2.
(
a
) Frequency distribution of plume height (H
p
) for the eruptions of Sakurajima during
2009-2019. (
b
) Distribution of accumulated tephra load (S
ac
) at stations within 20 km from the
Minamidake crater.

Atmosphere 2021,12, 331 18 of 21
The cumulative spatial distribution of the total deposit load from 2009 to 2019 around
the volcano is shown in Figure A2b based on data collected by Kagoshima prefectural
government [
33
]. The accumulated mass ranges between 30 kg/m
2
near the northwestern
coast and up to 250 kg/m2to the south of the vent.
Appendix B Tephra Load Observation Using the Disdrometer
A disdrometer is an instrument used to measure the particle size and settling velocity
of each particle that passes through the laser beam irradiated area. Here, tephra deposit load
was measured using a network of one-dimensional disdrometer Parsivel
2
(OTT;
Figure A3
),
which can automatically measure the diameter and settling velocity of pyroclastic particles
with a temporal resolution sufficient to measure a time series of tephra deposit load
(e.g., [
31
]). The disdrometer emits a laser beam with a wavelength of 650 nm at an area
30 mm wide from the transmitter and detects the range and duration of the laser beam
drop when the fallen particle blocks the laser beam at the receiver unit 180 mm away from
the transmitter unit as the voltage change, and then measures the size and settling velocity
of each particle. The number of particles in an effective total of 960 classes is recorded by
combining the particle size (30 classes from 0.25 to 26 mm) and settling velocity (32 classes
from 0 to 22.4 m/s).
Figure A3. A disdrometer under observation.
To detect tephra deposits in any direction, disdrometers were installed at 17 points
in all directions from the crater of Minamidake (Figure A4). The disdrometer can record
the number of particles detected at user-selected intervals from the previous recording
time, and we set the recording interval to 1 min to take into account the duration of the
vulcanian eruption, the accuracy of the arrival time calculated by the advection-diffusion
model, and the data volume.
To estimate the deposit load from the disdrometer observations the following empirical
equation is used:
Sobs =2.15
30
∑
i=1
32
∑
j=1
ρ0
ij
4
3πdi
23Nij
0.18 ×0.03 (A1)
ρ0
ij =maxρij , 500(A2)
where
ρij
is the effective density calculated as shown in Figure 5. Note that
ρij
’ in
Equation (A2)
is the lower limit of the effective density for each particle size and settling velocity class.
The correlation between the tephra deposit load calculated by Equation (A1) and the load
determined from the tephra deposit load in 59 events collected during the same period,

Atmosphere 2021,12, 331 19 of 21
from May 2017 to October 2019, is shown in Figure A5. The calculated values are estimated
to be about 0.1–2.3 times larger than the measured values.
Figure A4.
The disdrometer station network. Red circles are the disdrometer stations and the gray
circle AVOT is the strain station.
Figure A5.
Correlation between tephra deposit load obtained from deposit sampling and the dis-
drometer observations and Equation (A2).
Appendix C WRF Setup
Version 4 of the Weather Research and Forecasting (WRF) model [
21
] was used to pro-
vide the initialization meteorological data for Tephra4D. Although Tephra4D calculations
are carried out in an offline manner, a high temporal output in WRF was used to capture
the resolved variability of the wind field, as this has been shown to lead to an improvement
in tephra transport modeling [34].
The simulations were initialized using the Medium-Range Weather Forecasts (ECMWF)
reanalysis dataset (ERA5 [
35
]) (
∆
x 31 km, 137 vertical levels, hourly output [
35
]). In total
three one-way nested domains were used with horizontal grid spacing decreasing from
7.5 km (151
×
151 grid points;
∆
t = 30 s) to 1.5 km (301
×
301 grid points;
∆
t = 6 s) and
finally to 0.3 km (376
×
426 grid points;
∆
t = 0.167 s) for the innermost domain (Figure A6).
The same grid spacing is used for the Tephra4D simulations. The Shin-Hong scale-aware
Planetary Boundary Layer (PBL) scheme [
36
] was used to parametrize turbulent motion
in the innermost domain as at
∆
x = 300 m it falls within the “turbulence gray zone” [
37
].
Other than the PBL scheme, a full physics-parameterization set was used: hydrometeor

Atmosphere 2021,12, 331 20 of 21
microphysics [
38
], long- and short-wave radiation [
39
], a surface layer scheme [
40
], and
a land surface model [
41
]. The simulations were initialized 18 h before each eruption to
allow for model spin up time. In the vertical, 58 vertical levels were used, with vertical
grid spacing at 50 m from the surface up to 1 km asl, increasing to a spacing of 1 km near
the top of the domain at 50 hPa (≈21 km).
Figure A6. Placement of the WRF domains.
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