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).
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, Tephra2
PY, Tephra4D are described in
Table 1.
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 movement 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
LA (
hseg,
d) be the areal concentration of particles of settling velocity
vt arriving at point A on the ground from a plume at the segregation height
hseg, then the tephra deposit load at point A
SA can be expressed as the sum of the areal concentrations of particles as a function of the segregation height and terminal velocity:
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
LA(
hseg,
vt) at point A, (
xA,
yA) away from the center when the center reaches an altitude of point A, is obtained from the following equation:
where
M is the mass of the particle cluster with settling velocity
vt segregated from the altitude
hseg 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
t and empirically given by an expression proportional to the first power of
t when
t does not exceed a threshold
FTT (Fall Time Threshold) or an expression proportional to the 2.5 power of
t when
t exceeds
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),
σ2 is calculated as follows:
where
K and
C are 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 Tephra2
PY,
C is defined as follows:
The relationship between the radius of the plume
r and the elevation from the vent
h is empirically given as follows [
19]:
and in Tephra2 and Tephra2
PY the following relationship is also given [
3]:
This relationship is also followed in Tephra4D. Using Equations (3)–(6), the time between the departure at the vent and the segregation at altitude
hseg,
tup(
hseg), is obtained as follows:
Applying Equations (3), (5) and (6) to a vulcanian eruption at Sakurajima,
K was 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 m
2/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.
2.2. The Trajectory Calculation
In the trajectory calculation, we consider a Cartesian coordinate system with the eastward, 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.
In each step, after the particle starts to move, the particle goes straight in the hexahedron, 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 recursion formula:
Each symbol is described in
Table 2. The recursion formula about time is as follows:
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 Δ
tx, Δ
ty, Δ
t𝑧 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
h asl under the crater is assumed as follows:
where
hvent is the altitude of the vent asl. Assuming the density of fluid at 0 m asl as
ρa0, the density of fluid
ρa at the altitude
h is calculated following the hypsometric formula:
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].
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
vt (m/s), and particle diameter
d (mm). Assuming that the shape of the pyroclastic particle is a spheroid, the relationship between
ρp,
vt and
d is represented by the following set of equations connected by the drag coefficient
CD and the Reynolds number
Ra [
6]:
where
g is the gravitational acceleration,
ηa,
ρa are the viscosity and density of the surrounding fluid (i.e., the atmosphere in this study), and
F is the shape parameter of the particles. Substituting Equations (13) and (14) into Equation (12) we get:
Based on the diagram of the diameter and shape parameter
F of the pyroclastic particles at Stromboli volcano [
23],
F is calculated using the following equation:
Assuming that
vt and
d are 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
−5 Pa∙s, 1.205 kg/m
3, the settling velocity, effective particle density is calculated as shown in
Figure 5.
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.
2.5. Tephra Segregation Profile
Since TSP
M(
hseg,
vt), which is substituted in Equation (2), is highly arbitrary, it was constrained using the model of the distribution function. At an altitude interval of Δ
h, the mass of tephra with terminal velocity
vt at altitude
hseg discretized between the vent altitude
hvent to the plume altitude
hvent +
hplume is given by the following equation defining the function of TSP as
m(
h):
where
Mtotal(
vt) 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(
vt). We applied three distribution functions
m(
h) in this study.
1. Logarithmic Gauss distribution with top concentration;
2. Logarithmic Gauss distribution with bottom concentration;
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
hseg is restricted to the height from the vent to the plume top. As such, the sum of
M in Equation (17) is smaller than
Mtotal.