Lattice Gas Cellular Automaton to Model Volcanic Eruptions (NG51D-1670)Laura Sanchez? and Robert Shcherbakov?,†

?Department of Earth Sciences, University of Western Ontario, London, Ontario, N6A 5B7, Canada.†Department of Physics and Astronomy, University of Western Ontario, London, Ontario, N6A 3K7, Canada.

E-mails: lsanche@uwo.ca, rshcherb@uwo.ca

Abstract

Volcanic eruptions are the result of complex mechanisms that op-erate in a magma chamber within the crust. In a previous study, weshowed that the dynamics of eruptions on Earth are the same andare quite independent of the location and type of volcanism. Thegoal of this study is to test the universality of volcanism by design-ing a simple, general model to simulate processes occurring within amagma chamber. We aim at reproducing the threshold behavior thatoperates in the crust when pressure increase leads to an eruption.

To simulate volcanic eruptions, we propose a model based on lat-tice gas cellular automata (LGCA) rules, which have been provenefficient to simulate fluid flow behavior. This type cellular automa-ton is a discrete dynamical model in space and time, where the fluidis represented at the microscopic level by discrete particles. We startwith the simplest LGCA: the 2-dimensional HPP model (proposedin 1973 by Hardy et al.), which consists of a square lattice whereparticles interact with one another mimicking the fluid flow and con-serving mass and momentum. In this model, magma propagates bybreaking bounds within the crust. Once the magma reaches the topof the crust and the pressure threshold is exceeded, an eruption ora cascade of eruptions occur. We record the size of each event andthe number of time steps between consecutive events (or intereventtime). The model simulation results for a large number of realiza-tions are compared with observed data. The observations come fromeruption records of 13 individual volcanoes located around the worldas well as 11 groups of volcanoes located in various regions sur-rounded by different tectonic settings. From these, we computedthe frequency-size distribution of eruptions and the interevent timedistributions for a large number of active volcanoes on Earth. Thismodel allows us to study a large range of scales from the globalmagma movements to the interactions occurring at the pore level.Preliminary results show a good agreement between the model andthe data and this confirms the universal nature of volcanism.

1 Introduction

IMotivations:•To understand the general temporal structure of vol-

canic eruptions on Earth.

•To understand the physical mechanisms responsible forvolcanic eruptions.

2 Results from statistical analysis

•Data from 13 individual volcanoes located around theWorld and 11 groups of analog volcanoes (same geo-logical settings).

• Interevent time: time from the onset of one eruption tothe other.

•We computed the probability density function (PDF) ofinterevent time for each dataset:

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 61 0 - 9

1 0 - 8

1 0 - 7

1 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

A m b r y m A s a m a A s o E t n a G r i m s v o t n K i l a u e a M a u n a L o a M a r a p i M e r a p i P i t o n d e l a F o u r n a i s e S e m e r u T a u p o V i l l a r r i c a

P(∆t)

∆t [ d a y s ]

F i g u r e 3 : S a n c h e z a n d S h c h e r b a k o v ( 2 0 1 1 )

A l a s k a A l e u t i a n C e n t r a l A m e r i c a I c e l a n d I n d o n e s i a I t a l y J a p a n K a m c h a t k a N e w Z e a l a n d S o u t h A m e r i c a C a l d e r a s

Figure 1. Interoccurence time distribution in days (Sanchez and Shcherbakov,

2011 submitted). Data from Smithonian institution Global Volcanism Program

(Simkin and Siebert, 2000-).

•We rescale the distributions according to:

P(∆t) = τ f(∆tτ

)where τ is the mean interoccurence time of each distribution

•All the distributions collapse into a single curve that canbe described by the log-normal distribution:

f (x) =1

xσ√

2πexp

[−

ln(x)−µ2σ2

]where µ = −0.91±0.04 and σ = 1.32±0.03.

1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 1 0 1 1 0 21 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

A l a s k a g r o u pA l e u t i a n g r o u pC e n t r a l A m e r i c a g r o u pI c e l a n d g r o u p I n d o n e s i a g r o u p I t a l y g r o u pJ a p a n g r o u pK a m c h a t k a g r o u pN e w Z e a l a n d g r o u pS o u t h A m e r i c a g r o u p C a l d e r a s L o g - n o r m a l d i s t r i b u t i o n

A m b r y m A s a m a A s o E t n a G r i m s v o t n K i l a u e a M a u n a L o a M a r a p i M e r a p i P i t o n d e l a F o u r n a i s e S e m e r u T a u p o V i l l a r r i c a

� P(∆t

)

∆t / �

F i g u r e 4 : S a n c h e z a n d S h c h e r b a k o v ( 2 0 1 1 )Figure 2. Rescaled distributions of interevent times using the mean intervent

time as the scaling parameter. The solid curve is the fitted log-normal distribu-

tion (Sanchez and Shcherbakov, 2011 submitted).

•The mechanism responsible for volcanic eruptions oper-ates the same way for most volcanoes considered.

3 The Model

•Cellular Automata have been successfull to model com-plex behavior of physical systems. Several modelswere applied to volcanism: Pelletier (1992), Lahaie andGrasso (1998).

•Lattice Gas CA: for the simulation of fluid flow.•HPP model (Hardy et al., 1973): real gases or liquids

are modeled using discrete particles.

I The HPP model rules: Collision and Propagation on asquare lattice:•Rule for the propagation of particles for a configuration with 4 particles on a

site: no change after collision.

•Rule for the propagation of particles for a configuration with 3 particles on a

site: no change after collision.

•Rule for the propagation of particles for a configuration with 2 particles on a

site: head on collision, then propagation.

I The Eruption rules

• Left: The magma chamber (in red) and the crust (in brown). Right: Particles

within the crust. The particles in the oval are part of the eruption.

• In the magma chamber: site strength S = 0.• Initially, in the crust: site strength S = 1.•For magma to propagate within the crust, the site has to

be broken: S = 0.•Each site in the crsut has a damage and a healing capac-

ity: 0 ≤ D ≤ 1 and 0 ≤ H ≤ 1.

• If a particle is hitting a site at tn:

S (tn+1) = S (tn)−D• If no particle is hitting a site at tn:

S (tn+1) = S (tn) + H

I The Rules for eruption•Eruption when a particle reaches the top of the crust and

S = 0.•All the particles in the column under this site exit the

system during the eruption.•The eruption stops when no particles exit the system dur-

ing a time step.

4 Results

•Chamber size = 50×50, crustal size = 50×20I Small healing: H = 0.1 and H = 0.5

1 0 0 1 0 1 1 0 2

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

d a m a g e = 0 . 0 2 d a m a g e = 0 . 0 5 d a m a g e = 0 . 0 8 d a m a g e = 0 . 1

P(∆t)

∆t [ t i m e s t e p s ]

H e a l i n g = 0 . 1a )

1 0 - 1 1 0 0 1 0 11 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

d a m a g e = 0 . 0 2 d a m a g e = 0 . 0 5 d a m a g e = 0 . 0 8 d a m a g e = 0 . 1 0 L o g - n o r m a l d i s t r i b u t i o n

� P(∆t

)

∆t / �

H e a l i n g = 0 . 1b )

µ = - 0 . 3 1 0 6 � 0 . 0 0 8 1σ = 0 . 8 0 2 2 � 0 . 0 0 5 7

Figure 3. a) Pdf of interevent times for different values of D and H = 0.1.b) Rescaled densities using the mean intervent time, and log-normal fit.

1 0 0 1 0 1 1 0 21 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

d a m a g e = 0 . 0 2 d a m a g e = 0 . 1 0 d a m a g e = 0 . 4 0

P(∆t)

∆t [ t i m e s t e p s ]

H e a l i n g = 0 . 5a )

1 0 - 1 1 0 0 1 0 11 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

µ = - 0 . 2 6 3 1 � 0 . 0 0 3 2σ = 0 . 7 6 0 1 � 0 . 0 0 2 3

d a m a g e = 0 . 0 2 d a m a g e = 0 . 1 0 d a m a g e = 0 . 4 0 L o g - n o r m a l d i s t r i b u t i o n

� P(∆t

)

∆t / �

H e a l i n g = 0 . 5b )

Figure 4. a) Pdf, H = 0.5. b) Rescaled densities and log-normal fit.

I Large healing: H = 0.9 and H = 1.0

1 0 0 1 0 1 1 0 2 1 0 31 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

d a m a g e = 0 . 0 2 d a m a g e = 0 . 5 d a m a g e = 0 . 8

P(∆t)

∆t [ t i m e s t e p s ]

H e a l i n g = 0 . 9a )

1 0 - 1 1 0 0 1 0 11 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

µ = - 0 . 4 3 9 4 � 0 . 0 0 6 4 σ = 1 . 0 0 7 � 0 . 0 0 4 5λ = 1 . 0 4 9 � 0 . 0 0 8 5

d a m a g e = 0 . 0 2 d a m a g e = 0 . 5 0 d a m a g e = 0 . 8 0 L o g - n o r m a l d i s t r i b u t i o n E x p o n e n t i a l d i s t r i b u t i o n

� P(∆t

)

∆t / �

H e a l i n g = 0 . 9b )

Figure 5. a) Pdf, H = 0.9. b) Rescaled densities, exponential and log-normal fit.

1 0 0 1 0 1 1 0 2

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

d a m a g e = 0 . 6 0 d a m a g e = 0 . 8 0

P(∆t)

∆t [ t i m e s t e p s ]

H e a l i n g = 1 . 0a )

1 0 - 1 1 0 0 1 0 11 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

µ = - 0 . 4 3 8 3 � 0 . 0 0 2 5 σ = 1 . 0 0 5 � 0 . 0 0 1 7λ = 1 . 0 6 1 � 0 . 0 0 3 5

d a m a g e = 0 . 6 0 d a m a g e = 0 . 8 0 L o g - n o r m a l d i s t r i b u t i o n E x p o n e n t i a l d i s t r i b u t i o n

� P(∆t

)

∆t / �

H e a l i n g = 1 . 0b )

Figure 6. a) Pdf, H = 1.0. b) Rescaled densities, exponential fit and log-normalfit.

1 0 0 1 0 1 1 0 21 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

H e a l i n g = 0 . 1 d a m a g e = 0 . 0 2 H e a l i n g = 0 . 1 d a m a g e = 0 . 0 5 H e a l i n g = 0 . 1 d a m a g e = 0 . 0 8 H e a l i n g = 0 . 5 d a m a g e = 0 . 0 2 H e a l i n g = 0 . 5 d a m a g e = 0 . 1 0 H e a l i n g = 0 . 5 d a m a g e = 0 . 4 0

P(∆t)

∆t [ t i m e s t e p s ]

Figure 7. Probability density functions of interevent times for H = 0.1 andH = 0.5.

1 0 - 1 1 0 0 1 0 11 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

µ = - 0 . 2 8 1 6 � 0 . 0 0 3 1 σ = 0 . 7 9 7 � 0 . 0 0 2 2

H e a l i n g = 0 . 1 d a m a g e = 0 . 0 2 H e a l i n g = 0 . 1 d a m a g e = 0 . 0 5 H e a l i n g = 0 . 1 d a m a g e = 0 . 0 8 H e a l i n g = 0 . 5 d a m a g e = 0 . 0 2 H e a l i n g = 0 . 5 d a m a g e = 0 . 1 0 H e a l i n g = 0 . 5 d a m a g e = 0 . 4 0 H e a l i n g = 0 . 5 d a m a g e = 0 . 5 0 L o g - n o r m a l d i s t r i b u t i o n

� P(∆t

)

∆t / �

Figure 8. Rescaled distributions using the mean interevent time as a scalingparameter and fitted log-normal distribution.

I Size distribution of eruptions

1 0 1 1 0 21 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

µ = - 1 . 6 9 2 � 0 . 0 1 0 2σ = 0 . 9 7 1 � 0 . 0 0 7 2

H e a l i n g = 0 . 1 d a m a g e = 0 . 0 2 L o g - n o r m a l d i s t r i b u t i o n

P(VEI)

V E I [ n u m b e r o f p a r t i c l e s ]

Figure 9. Probability density function of the eruption sizes for H = 0.1 andD = 0.02. The solid curve is the fitted log-normal distribution.

5 Conclusion

I Preliminary results are in accordance with real data.I The interevent times distribution modeled for small Hvalues reproduce qualitatively the real data distributions.I The log-normal distribution is playing an importantrole in describing the temporal structure of volcanic erup-tions.

References

•Hardy J., Pomeau Y., and de Pazzis O., J. Math. Phys., 14, 12, 1746-1759

(1973).

• Lahaie F. and Grasso J.R., J. Geophys. Res., 103, B5, 9637-964 (1998).

• Pelletier J.D., J. Geophys. Res., 104, B7, 15425-15438, (1999).

• Sanchez L., Shcherbakov R., (2011), submitted.

• Siebert L. and Simkin T., Global volcanism program digital informa-

tion series, GVP-3, Smithonian institution,(http://www.volcano.si.edu/world/)

(2000).

I Acknowledegments This work has been supported by NSERC Discovery

grant 355632-2008 and UWO ADF grant R44203A03