Pyroclastic avalanches are a type of granular flow generated at active
volcanoes by different mechanisms, including the collapse of steep
pyroclastic deposits (e.g., scoria and ash cones), fountaining during
moderately explosive eruptions, and crumbling and gravitational collapse of
lava domes. They represent end-members of gravity-driven pyroclastic flows
characterized by relatively small volumes (less than about 1 Mm

Pyroclastic avalanches are rapid flows of pyroclastic material (volcanic ash,
lapilli, pumices and scoriae) that propagate down volcanic slopes under
the effect of gravity. They share many phenomenological features with other
natural granular avalanches (such as landslides and rock and snow
avalanches) and they pose similar modeling, monitoring and risk mitigation
challenges

Despite the fact that the term pyroclastic avalanche is not widely used among
volcanologists, who often adopt the term pyroclastic density current (PDC),
there are reasons to prefer the former in some specific cases. The term
avalanche derives from the old french

We will refer to pyroclastic avalanches in this work for pyroclastic flows
that (1) remain confined within the volcano slopes, (2) show evidence of a
dense basal granular flow and (3) are controlled by topography (i.e., they
mostly move in the direction of the maximum slope). Such conditions generally
reduce the applicability of this category to relatively small flows (less
than about 1 million m

As in classical fluid dynamics, even more so for granular fluids, the choice
is between continuum and discrete field representation

Despite these simplifying hypotheses, several difficulties arise in granular
avalanche depth-averaged models. On the one hand, terrain-following coordinates
are often used to express depth-averaged transport equations. However, on
3-D rough surfaces, they need to be corrected with curvature terms, which
introduce problems with irregular topographies, cliffs, obstacles and high
curvatures

On the other hand, from a physical point of view, the description of the
depth-averaged rheology of the granular fluid was revealed to be problematic for
strongly stratified and nonhomogeneous flows

In addition to the latter, some additional difficulties arise from the
numerical solution to the conservation equations: despite the fact that numerical
methods
based on conservative, approximate Riemann solvers are robust and well tested

In this paper, we present and show verification tests of the new IMEX_SfloW2D (which stands for Implicit–Explicit Shallow Water flow) numerical model for shallow granular avalanches that we designed to address most of the above difficulties. In particular, the model is formulated in a geographical (absolute) coordinate system so that it is possible to include non-hydrostatic terms arising from steep topographic slopes, rapid topographic changes (e.g., local curvature) or from more accurate approximation of the vertical momentum equations. In this version of the model, however, non-hydrostatic terms are neglected. The model can deal with different initial and boundary conditions, but its first aim is to treat gravitational flows over topographies described as digital elevation models (DEMs) in the ESRI ASCII format. The same format is used for the output of the model so that it can be handled very easily with GIS software.

Several different geophysical flow depth-averaged models already exist, which
are able to simulate pyroclastic avalanches on DEMs. Among them, two of the
most well-established models in the volcanological community are VolcFlow and
Titan2D. VolcFlow is a MATLAB finite-difference Eulerian code based on an
explicit upwind or double-upwind discretization. Recently, A two-layer
depth-averaged version for both the dilute and the concentrated parts of
pyroclastic currents has been implemented

Numerically, the first and most relevant advancement in IMEX_SfloW2D is represented by the implicit treatment of the source terms in the transport equations, which avoids most problems related to the stopping of the flow, especially when dealing with strongly nonlinear rheologies. Any rheological model can in principle be implemented, including formulations not dependent on velocity, as purely frictional or plastic rheologies without the need for introducing an artificial numerical viscosity. The model is indeed discretized in time with an explicit–implicit Runge–Kutta method whereby the hyperbolic part and the source term associated with topography slope are solved explicitly, while other terms (friction) are treated implicitly. The finite-volume solver for the hyperbolic part of the system is based on the Kurganov and Petrova (2007) semi-discrete central-upwind scheme and it is not tied to the knowledge of the eigenstructure of the system of equations. The implicit part is solved with a Newton–Raphson method whereby the elements of the Jacobian on the nonlinear system are evaluated numerically with a complex-step derivative technique. This automatic procedure allows for the use of different formulations of the friction term without the need for major modifications of the code. In particular, the Voellmy–Salm empirical model is implemented in the present version.

The FORTRAN90 code can be freely downloaded and it is designed in a way that users can simply use it without any intervention or they can easily modify it by adding new transport and/or constitutive equations.

In this section we present the governing equations based on the shallow water approximation. We omit their derivation that comes from the manipulation of the mass conservation law and Newton's second equation of motion.

The model we use for the flow evolution is described by the Saint-Venant
equations

With these assumptions, and without considering frictional forces, the 2-D
inviscid depth-averaged equations in differential form can be written in the
following way:

The first equation represents the conservation of mass (or volume because of the
constant density), while the other two equations describe the conservation of
momentum in the

List of model variables with notation and units.

As stated above, the variables of the model are

If we introduce the vector of conservative variables

To properly model shallow pyroclastic avalanches, we have to modify the
classic Saint-Venant equations by introducing an additional source term

The terms we consider appear only in the momentum equations (

IMEX_SfloW2D is based on a finite-volume central-upwind scheme in space and
on an implicit–explicit Runge–Kutta scheme for the discretization in time. The
main purpose of the code is to run simulations on colocated grids derived
from DEMs, and for this reason the standard input files defining the
topography are raster files in the ESRI ASCII format, defining a uniform grid
of equally sized square pixels whose values (in our case representing the
terrain elevation above sea level) are arranged in rows and columns. The
procedure to define the elevation values at the face centers and cell centers
of the computational grid is represented in Fig.

Computational grids. The colored pixels represent the elevation values of the original DEM. The lines define the edges of the IMEX-SfloW2D computational cells. The elevation values at the centers (filled squares), faces (no-fill squares) and corners (no-fill circles) of the computational cells are obtained by interpolating the pixel values associated with their centers (filled circles).

The finite-volume method adopted for IMEX_SfloW2D is based on the
semi-discrete central-upwind scheme introduced in

The choice of the variables to reconstruct at the interface is fundamental
for the stability of the numerical scheme. The homogeneous system associated
with Eq. (

For the reconstruction procedure based on the physical variables, we
introduce the notation

During the reconstruction step, particular care should be taken to
avoid unrealistic values of the physical variables, such as negative flow
thickness or velocities that are too large. For this reason, in the case that one of the
reconstructed interface values of

Finally, once the physical variables are reconstructed at the interfaces, the
numerical fluxes in the

Following

The semi-discrete system of Eq. (

The family of IMEX methods

An IMEX Runge–Kutta with

Following

1-D Riemann problem with no friction. Numerical solution at three
different times: 0 s

In this section we present numerical tests aimed at demonstrating the
mathematical accuracy of the numerical model results

the capability to manage the propagation of discontinuities;

the potential to deal with complex and steep topographies and dry–wet interfaces; and

the ability of the granular avalanche to stop, thereby achieving the expected steady state.

All the numerical tests presented here are available on the Wiki page of the
code (

The first example is a 1-D test for a Riemann problem with a discontinuous
topography, as presented in

The domain is the interval

The gravitational constant is

The initial solution (Fig.

Numerical solution of a 1-D Riemann problem with friction at four different times. The solid blue line represents the topography, while the solid green line represents the free surface of the wet region.

This example is a 1-D test for a system with friction, as presented in
Kurganov and Petrova (2007). As in the previous test, an initial
discontinuity is present, but this time representing the interface from a
“wet” (presence of flow) and a “dry” (no flow or zero thickness) region,
with the terminology borrowed from the common use of Saint-Venant equations
in hydrology. Thus, there is an additional numerical difficulty involving the
capability of the numerical solver to propagate these discontinuities without
creating regions with negative flow thickness. For this problem the bottom
topography presents both smooth regions and a step, and it is defined as
follows:

Numerical solution of a 1-D problem with Voellmy–Salm friction at
three different times. In

This example is a 1-D test for the Voellmy–Salm rheology, with a pile of
material initially at rest released on a constant slope topography. This test
is aimed at checking if the model is able to preserve an initial steady
condition, when the tangent of the pile free surface slope is smaller than
the Coulomb friction coefficient,

For this test, the domain is the interval

We present the results for a numerical simulation with a topography with a
constant slope of 13

The numerical solution at three different times (

Numerical simulation of two-dimensional pyroclastic avalanche with Voellmy–Salm friction. The contour plots on the bottom plane of each panel represent constant values of the thickness of the flow, whose free surface is represented in blue. The outermost contour on the bottom plane corresponds to a thickness of 0.06 m, and thus the thinner portion of the flow is not represented by the contour plot. A visual comparison between the two bottom plots highlights the fact that a steady condition has been reached.

This test extends the simulation with a Voellmy–Salm rheological model
presented in the previous section from one to two dimensions, with an example
of an avalanche of finite granular mass sliding down an inclined plane
merging continuously into a horizontal one. The initial conditions and the
topography of this tutorial are the same as in Example 4.1 from

The numerical solution at four different times (

Map of the avalanche deposit from the February 2014 pyroclastic avalanche and the lava flow.

As shown by the plots presented in Fig.

Results of IMEX_SfloW2D numerical simulations of a pyroclastic avalanche overlapped on the
hill-shaded relief of the Etna summit and the boundaries of the 11 February 2014 event.
Numerical parameters as follows:

On 11 February 2014, a hot pyroclastic avalanche was generated at the New
Southeast Crater (NSEC) of Etna, triggered by the instability and collapse
of its eastern flank where several vents had been actively effusing lava flows
towards Valle del Bove since 22 January. The avalanche propagation was
recorded by the INGV (Istituto Nazionale di Geofisica e Vulcanologia) monitoring
system and, in particular, it was filmed by the thermal IR camera from Monte
Cagliato, located on the east slope of the Valle del Bove at about 7 km from
the NSEC, and the Catania CUAD visible camera (ECV) about 26 km south of the
summit. A 500 m wide avalanche front propagated about 2.3 km along the
steep slopes of the Valle del Bove before stopping at the break in slope at
the valley bottom. At the same time, a voluminous buoyant ash cloud was
generated by elutriation of the finest ash from the avalanche and rapidly
dispersed in the north-northeast direction by an intense wind. The event is
accurately reported by

IMEX_SfloW2D numerical simulations have been performed over the 2014 digital
elevation model of Mount Etna

The avalanche rheological parameters have been varied in ranges consistent
with previous studies of geophysical granular avalanches

The misfit between the simulated and observed flow path is due to the
presence of the lava flow depicted in Fig.

A thorough discussion about the optimal choice of the rheological model and parameters for pyroclastic avalanches would require an extensive comparison with similar phenomena that occurred at Etna (few of which have been documented so far) and at other analogous volcanoes, for which more accurate measurements will be needed in the future to achieve a better calibration of the model. This, is however, clearly beyond the scope of the present work.

We have presented the physical formulation, numerical solution strategy and
verification tests of the new IMEX_SfloW2D numerical model for shallow
granular avalanches. The numerical code is available open-source and freely
downloadable from a GIT repository, where the users can also find the
documentation and example tests described in this paper. The main features of
the new model make it suited for research and application to geophysical
granular avalanches, in particular the following.

The flexible discretization and numerical solution algorithm (not tied to knowledge of the eigenstructure of the system of equations) allows for the easy implementation of new transport equations.

The formulation in Cartesian geographical coordinates is suited for running on digital surface models (read in standard ESRI ASCII grid format) and for the integration of non-hydrostatic terms, even on steep slopes.

The conservative and positivity-preserving numerical scheme allows for a robust and accurate tracking of 1-D and 2-D discontinuities, including wet–dry interfaces and flow fronts.

The implicit coupling of nonlinear rheology terms allows for the simulation of steady-state equilibrium solutions and, in particular, favors flow stopping without the need for any ad hoc empirical criteria.

The numerical procedure to evaluate the Jacobian of the nonlinear system (based on a complex-step derivative technique) allows for an easy implementation and testing of new rheological models for complex geophysical granular avalanches.

The numerical code, benchmark tests and documentation are
available at

MdMV has developed the numerical algorithm and implemented the 1-D model and numerical tests. TEO has contributed to the model formulation and application in the context of volcanological applications. GL has implemented the numerical scheme and algorithm in 2-D. AA has tested the model and helped with the comparison to similar depth-averaged models in volcanology.

The authors declare that they have no conflict of interest.

The work has been supported by the Italian Department of Civil Protection, INGV-DPC agreement B2 2016, task D1. We warmly thank Daniele Andronico, Emanuela De Beni and Boris Behncke for useful discussions about the Etna 2014 event and for providing field data and the digital elevation model. TEO would like to thank Perry Bartelt and Betty Sovilla (SLF, Switzerland) for stimulating discussions on granular avalanches and density currents. The authors are also grateful to Olivier Roche and Karim Kelfoun for their careful review and useful and constructive comments. Mattia de' Michieli Vitturi and Giacomo Lari gratefully acknowledge Andrea Milani, whose contribution to the career and numerical work of the two authors was of great significance.Edited by: Simone Marras Reviewed by: Karim Kelfoun and Olivier Roche