SCIARA γ2: An improved cellular automata model for lava flows and applications to the 2002 Etnean crisis
Introduction
Lava flows represent one of the greatest dangers for people security and involve invasion of land and property. Lava flow simulation could abate this hazard by forecasting lava paths and evaluating the effects of human interventions such as the construction of embankments or channels. Nonetheless, lava flows are complex phenomena and need, in general, sophisticated modelling tools. In fact, the major difficulty in modelling lava flows arises since derived equations (i.e. conservation, state and constitutive equations) must also satisfy additional variables to describe the physical properties of lava and its environment. For instance, because of cooling and crystallization, lava may change from a viscous fluid to a brittle solid during emplacement. Consequently, it is extremely difficult to characterize the resistance of lava to motion (Kilburn and Luongo, 1993). This paper deals with a different approach, modelling flow field growth with Cellular Automata techniques.
Cellular Automata (CA) are one of the first Parallel Computing models (von Neumann, 1966); they capture the peculiar characteristics of systems, whose global evolution may be described on the basis of local interactions of their constituent parts (i.e. locality property).
A homogeneous CA (Worsch, 1999) can be considered as a d-dimensional space, the cellular space, partitioned into regular cells of uniform size, each one embedding an identical finite automaton, the elementary automaton (ea). Input for each ea is given by the states of the ea in the neighbouring cells, where neighbourhood conditions are determined by a pattern, which is invariant in time and constant over the cells. At time (step 0), ea are in arbitrary states and the CA evolves changing the state of all ea simultaneously at discrete times, according to the transition function of the ea.
Complex phenomena modelled by classical CA involve an ea with few states (usually no more than two dozens) and a simple transition function, easily specified by a lookup table (Toffoli and Margolus, 1987).
Fluid dynamics is an important field of CA application: lattice-gas automata models (Frisch et al., 1990) and lattice-Boltzmann models (Succi, 2001) were introduced for describing the motion and collision of “fluid particles” on a discrete space/time. It was shown that such models could simulate fluid-dynamical properties; the continuum limit of these models leads to the Navier–Stokes equations.
This CA approach does not permit to make velocity explicit in the local context of the cell: i.e., an amount moves from the central cell to an adjacent cell in a CA step (which is a constant time), implying a constant “velocity”. Nevertheless, velocities can be deduced by analysing the global behaviour of the system in time and space. In such models, the flow velocity is an emergent property. It can be deduced implicitly by averaging quantities on space (i.e. considering clusters of cells) or by averaging quantities on time (e.g. considering the average velocity of the advancing flow front in a sequence of Cellular Automaton steps).
Many complex macroscopic fluid-dynamical phenomena, which own the same locality property of CA, the surface flows, like lava flows, seem difficult to be modelled in these CA frames, because they take place on a large space scale and need practically a macroscopic level of description that involves the management of a large amount of data, e.g., the morphological data.
Simulation by Cellular Interactive Automata of the Rheology of Aetnean lava flows (SCIARA) γ2 adopts and extends some mechanisms (Di Gregorio and Serra, 1999), which permit to define the macroscopic phenomenon of lava flow in terms of CA formalism. In particular, the flows are characterized by a mass centre position (inside the cell), that changes according to the velocity and produces an outflow, when position moves to a neighbouring cell.
The second section considers some CA empirical criteria for modelling complex macroscopic surface flows and the introduction of the explicit velocity in the previous methods; the third section presents the main specifications of the release γ2 of SCIARA; the transition function with its innovations is described in the fourth section; the fifth section shows a result of the simulations of some of the numerous lava flows, which occurred in the 2002 Etnean eruption, Sicily, Italy1 together with a comparison with application results of the previous SCIARA release; finally, some conclusions are reported at the end.
Section snippets
Some criteria for modelling surface flows with CA
The criteria here reported are empirical and may be applied only to particular classes of macroscopic phenomena, involving surface flows. The recipes that are here presented have to be adapted to the special features of the phenomenon. However, in general, they do not guarantee by themselves success in the simulation. In fact, only when the model is validated on real cases of a certain typology, the model applications in similar conditions are reliable. SCIARA γ2 was validated on recent
The problem of modelling lava flows
Navier–Stokes equations represent a good, but perhaps not completely exhaustive approach in terms of systems of differential equations for accounting for the physical behaviour of lava flows (McBirney and Murase, 1984). Analytical solutions to these differential equations which govern debris flows, are a hopeless challenge, except for few simple, not realistic, cases. The possibility to successfully apply numerical methods for the solution of differential equations have been elevated
The SCIARA γ2 transition function
The “elementary” processes are specified in terms of equations, where names of parameters and substates are derived by the lower indexes specified in Section 3.2: the parameter pname is indicated as name; qname and n_qname indicate, respectively, the value of substate Qname and the new value; when the specification of the index i of a neighbourhood cell is necessary, the notation qname[i] is adopted; an index pair (i,j) is needed for the flows: qFname[i,j] means the value of the flow substate
Simulations of the lava flows of 2002
The event which occurred in the autumn of 2002 at Mount Etna (Sicily) was chosen for validating SCIARA γ2. Two major eruptions have characterized this event. The first, started on the NE flank of the volcano, with lava generated by a fracture between 2500 and 2350 m a.s.l., near the 1809 fracture and pointed towards the town of Linguaglossa (Fig. 4). The Linguaglossa forest and numerous tourist facilities were destroyed by the lava flow in the days following the eruption. Fortunately, after 8
Conclusions
SCIARA γ2 represents a quality leap in modelling lava flows with CA. The attribution of the velocity to blocks, which are identified by barycentre co-ordinates and move inside a neighbourhood, permits to overcome the limits of the previous versions of SCIARA, where the velocity was an emergent property of the system.
We do not need to univocally fix the CA clock in the validation phase on the basis of results of the simulations as in the previous versions of SCIARA, but we may choose the two
Acknowledgements
The authors are very grateful to Dr. Sonia Calvari of the “Istituto Nazionale di Geofisica e Vulcanologia” of Catania (Italy) for the logistic support and useful information on the 2002 Mt Etna event. We are grateful to Hideaki “Hirdy” Miyamoto for the precious suggestions he made in the revision phase of the paper and to the second anonymous reviewer.
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