Eruption source parameters (ESP) characterizing volcanic eruption plumes are
crucial inputs for atmospheric tephra dispersal models, used for hazard
assessment and risk mitigation. We present FPLUME-1.0, a steady-state 1-D
(one-dimensional) cross-section-averaged eruption column model based on the
buoyant plume theory (BPT). The model accounts for plume bending by wind, entrainment of
ambient moisture, effects of water phase changes, particle fallout and
re-entrainment, a new parameterization for the air entrainment coefficients
and a model for wet aggregation of ash particles in the presence of liquid water
or ice. In the occurrence of wet aggregation, the model predicts an

Volcanic plumes

Sketch of an axisymmetric volcanic plume raising
in a wind
profile. Three different regions (jet thrust, convective thrust and
umbrella) are indicated, with the convective region reaching a
height

Quantitative observations and models of volcanic plumes are essential to
provide realistic source terms to atmospheric dispersal models, aimed at
simulating atmospheric tephra transport and/or the resulting fallout deposit

Many plume models based on the BPT have been proposed after the seminal
studies of

Here we present FPLUME-1.0, a steady-state 1-D cross-section-averaged plume
model, which accounts for plume bending, entrainment of ambient moisture,
effects of water phase changes on the energy budget, particle fallout and
re-entrainment by turbulent eddies, variable entrainment coefficients fitted
from experiments and particle aggregation in presence of liquid water or ice
that depends on plume dynamics, particle properties, and amount of liquid
water and ice existing in the plume. The modelling of aggregation in the
plume, proposed here for the first time, allows our model to predict an

We consider a volcanic plume as a multiphase mixture of volatiles, suspended
particles (tephra) and entrained ambient air. For simplicity, water (in
vapour, liquid or ice phase) is assumed the only volatile species, being
either of magmatic origin or incorporated through the ingestion of moist
ambient air. Erupted tephra particles can form by magma fragmentation or by
erosion of the volcanic conduit, and can vary notably in size, shape and
density. For historical reasons, field volcanologists describe the continuous
spectrum of particle sizes in terms of the dimensionless

List of Latin symbols. Quantities with a hat
denote bulk (top-hat averaged) quantities. Throughout the text, the subindex
o (e.g.

List of greek symbols. Quantities with a hat denote bulk (top-hat averaged) quantities.

The steady-state cross-section-averaged governing equations for axisymmetric
plume motion in a turbulent wind (see Fig.

The equations above derive from conservation principles assuming axial
(stream-wise) symmetry and considering bulk quantities integrated over a
plume cross section using a top-hat profile in which a generic quantity

Assuming an homogeneous mixture, the bulk density

The model uses a pseudo-gas assumption considering that the mixture of air
and water vapour behaves as an ideal gas:

For the particle re-entrainment parameter

Turbulent entrainment of ambient air plays a key role on the dynamics of jets
and buoyant plumes. In the basal region of volcanic columns, the rate of
entrainment dictates whether the volcanic jet enters into a collapse regime
by exhaustion of momentum before the mixture becomes positively buoyant, or
whether it evolves into a convective regime reaching much higher altitudes.
Early laboratory experiments

Entrainment functions

Constants defining the entrainment
functions for jets and plumes following the formulation introduced by

The umbrella region is defined as the upper region of the plume, from about
the NBL to the top of the column. This region can be dominated by fountaining
processes of the eruptive mixture that reaches the top of the column,
dissipating the excess of momentum at the NBL, and then collapsing as a
gravity current

In the umbrella region (from the NBL to the top of the column), we neglect
air entrainment and assume that the mixture is homogeneous, i.e. the
content of air, water vapour, liquid water, ice and total mass of particles
do not vary with

Finally, assuming that the kinetic energy of the mixture is converted to
potential energy, the vertical velocity is approximated to decrease as the
square root of the distance from the NBL:

Particle aggregation can occur inside the column or in the ash cloud during
subsequent atmospheric dispersion

Entrainment coefficients

The total particle decay per unit volume and time

The total number of particles per unit of volume available for aggregation is
related to particle class mass concentration at each section of the plume

We solve the model equations using FPLUME-1.0, a code written in FORTRAN90
that uses the lsode library

Model outputs include a text file with the results for each eruption phase
giving values of all computed variables (e.g.

At each section of the plume, determine the water vapour condensation
or deposition conditions depending on

In case of saturation or deposition, compute the class-averaged
sticking efficiency

Estimate the total number of particles per unit of volume
available for aggregation

Compute the integrated aggregation kernels using Eq. (

Compute the total particle decay per unit volume and time

Compute the number of particles of diameter

Compute class particle decay

Finally, compute the mass sink term for each aggregating
class

As we mentioned above, here we apply FPLUME to two eruptions relatively well
characterized by previous studies. In particular we consider the strong plume
formed during 4 April 1982 by El Chichón 1982 eruption

El Chichón volcano reawakened in 1982 with three significant Plinian
episodes occurring during March 29th (phase A) and April 4th (phases B and
C). Here we focus on the second major event, starting at 01:35 UTC on April
4th and lasting nearly 4.5 h

Input values for the El Chichón
Phase-B simulation. Values for specific heat of water vapour, liquid water,
ice, pyroclasts and air at constant pressure are assigned to defaults of
1900, 4200, 2000, 1600 and
1000

We use this test case to verify whether FPLUME can reproduce results from
these previous studies and the results of our aggregation model are, in this
case, consistent with those of

Dependency of fractal exponent

El Chichón 1982 phase-B simulation. Total
mass fraction of aggregates (red line) and total mass fraction of aggregates
with respect to fines (blue line), depending on the fractal exponent

A clear advantage of a physical aggregation model of ash particles inside the
eruption column, with respect to an empirical parameterization like

The infamous April–May 2010 Eyjafjallajökull eruption, that disrupted the
European North Atlantic region airspace

Results of the aggregation model in FPLUME
for El Chichón 1982 phase-B simulation. Green bars show the original TGSD
from

Original grain size distribution from ground
data and MSG-SEVIRI retrievals (green) and distribution modified by
aggregation (red). Results are for 6 May 30 min averaged. Figure reproduced
from

Preliminary simulations using time-averaged plume heights of
3.5–4.5

FPLUME input values for the 6 May
Eyjafjallajökull simulation. Values for specific heats of water vapour,
liquid water, ice, pyroclasts and air at constant pressure are assigned to
defaults of 1900, 4200, 2000, 1600 and
1000

Atmospheric profiles extracted form
ERA-Interim re-analysis data set at Eyjafjallajökull vent location for 6
May 2010 at 12:00 UTC.

FPLUME aggregation model results for
Eyjafjallajökull 6 May phase. Total mass fraction of aggregates (in %)
versus mass
flow rate (in

Grain size distribution predicted by
the wet aggregation
model for Eyjafjallajökull 6 May phase for a column height of
6.5

Input values for FPLUME are summarized in Table

We presented FPLUME, a 1-D cross-section-averaged volcanic plume model based
on the BPT that accounts for plume bending by wind, entrainment of ambient
moisture, effects of water phase changes, particle fallout and
re-entrainment, a new parameterization for the air entrainment coefficients
and an ash wet aggregation model based on

The code FPLUME-1.0 is available under request for research purposes.

Denoting with

Consider a particle grain size distribution discretized in

Adding the contribution of all bins, this yields to

This work was partially supported by the MED-SUV Project funded by the European Union (FP7 grant agreement no. 308665). We acknowledge C. Bonadonna for providing grain size data for the Eyjafjallajökull test case. We thank T. Esposito Ongaro and two anonymous reviewer for their constructive comments.Edited by: R. Sander